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AP Physics - Magnetism

We’re going to look at magnets for a bit. The Physics Kahuna is absolutely convinced that all of his advanced students no exactly what a magnet is. No mystery here, you have got to be familiar with magnets. First of all, magnets are cool. You’ve probably had a magnet or two in your young life. There’s something wonderful about the way they can defy gravity and exert forces on things over distance. The Physics Kahuna would be willing to bet several shekels that you did a magnet experiment or two in grade school. Typical kind of thing would be to try and discover what kind of materials are attracted to magnets; steel screw – yes, wooden toothpick – no, penny – no, quarter – no, washer – yes, paperclip – yes, and so on.

If your elementary teachers were really good, you did the experiment where you put a piece of paper over a magnet and then sprinkled iron filings all around the magnet. The filings line up and make very interesting patterns.

Perhaps you learned to make a compass from a needle.

It turns out that magnets and magnetism are extremely important in modern life. Electric motors, TV’s, computers, electric generators, locomotives, CD players, all depend upon . . . well, you get the idea.

The first person to write about and study magnets was Thales about 2600 years ago. This is the same guy who looked into static electricity with the amber, remember? Thales found that rocks from a town called Magnesia could attract bits of iron. He called the things, "ho mangetes lithos" which means "the Magnesian rock". This is how the magnet got its name, from good old Magnesia (don’t confuse magnets from magnesia with Milk of Magnesia, which is a laxative).

Here’s your basic important circumstance --- magnetism and electricity are very closely connected – they are the two sides of a single coin, electromagnetism. So it’s interesting that one dude, Thales, was the first to study both of these two phenomena.

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Magnet Basics: Magnetism has a lot of similarities to electricity. Electricity involves two charges, positive and negative. Magnets have two poles - the north pole and south pole. Fundamental rule for magnets:

Like poles repel, unlike poles attract.

All magnets have these two poles. If you cut a magnet in half, the two new, smaller magnets will each have two poles. If you cut these halves into two more pieces, each of the new magnets will also have two poles. And so on. You can never slice a magnet in half and get only one pole. One of the interesting questions that modern physicists play around with is whether a magnet with a single pole can exist. Such a thing (which has never been discovered) is known as a monopole. Win yourself one of them Nobel Prizes in Physics get you one of them old monopoles.

Magnets exert forces on other magnets. They also can interact with other materials. The important interaction is the way they act with materials classed as ferromagnetic. These materials are strongly attracted to magnets. Ferromagnetic materials include the following elements: iron, cobalt, nickel, gadolinium, and dysprosium. Materials made with these elements (or compounds of these elements) are not only attracted to magnets, they can be magnetized and turned into magnets themselves.

Diamagnetic materials are weakly repelled by magnets. Many common materials are diamagnetic: water, glass, copper, graphite, salt, lead, rubber, diamond, wood, and many plastics for example.

Paramagnetic materials are weakly attracted to magnets. Examples: aluminum, oxygen, sodium, platinum, and uranium.

Magnetic Fields: Just as electric charges are surrounded by an electric field, so too are magnets surrounded by a magnetic field. We can even draw lines of force around the magnet to show the direction of the field and its strength. These are called magnetic lines of force or sometimes you see them referred to as magnetic lines of flux.

Here are some characteristics of a magnetic field:

The lines of flux travel through the magnet They leave the magnet at the north pole. They travel through the air in a curve. The lines enter the magnet at the south pole. A line tangent to any point on a line of flux shows the direction of the field – which is

the direction of the force that would be exerted on a north pole. Where the lines are close together the field is the strongest. The direction of the field is NORTH to SOUTH. The arrows point away from the

north pole and towards the south pole.

In the drawing below you can see some of the lines of force of a bar magnet. Three points are located on the lines of force and the corresponding forces that would be exerted at each point are shown. The force is always tangent to the line of force.

The direction of the field is shown by the direction of the arrows.

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Below are some of the lines of force between two magnets. The drawing on the left show the lines of force when two opposite poles face one another. The drawing on the right show the lines of force for two unlike poles.

The symbol for the magnetic field is B. The most common unit for the magnetic field is the Tesla (T). Other units can be used as well such as the gauss (G) and the Weber (Wb).

A Tesla is a newton per meter ampere:

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Magnetic & Electrostatic Forces: There are many similarities between magnetic and electrostatic fields. There are also a few differences.

Both obey an inverse square law (just like gravity does).

They can both be attractive or repulsive.

The primary difference between them is that the electrostatic charge can be a point charge, but magnets must always have a north and south pole.

Forces and Fields: Place a charged up balloon in a magnetic field – nothing happens. Magnetic fields don’t affect stationary charges. But a moving charge, well, that’s a whole different thing. A moving charge traveling through a magnetic field will experience a FORCE. The force exerted will be perpendicular to the motion of the charge and perpendicular to the direction of the field. The result of the force is to cause a deflection of the charged particle. It gets pushed to the side.

The equation for the force exerted on a moving charge by a magnetic field is:

FB is the magnetic force, B is the magnetic field in Tesla’s, q is the charge, v is the velocity of the charged particle, and is the angle between the velocity direction and the direction of the magnetic field.

You will have this equation available for your use on the dreaded AP Physics Test.

The force on the charged particle is at a maximum when the velocity is perpendicular to the magnetic field. Note that if the velocity is in the direction of the magnetic field, the magnetic force will be zero.

The force is always perpendicular to the velocity and the magnetic field, B. This is shown in the drawing below. At the center is a particle that has a charge q. The direction of the magnetic field B is to the right. The particle’s direction is out of the sheet yer a lookin’ at. This is v. Therefore the force must be directed up.

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To quickly figure out the direction of the force acting on the charged particle we can use the right hand rule.

Here’s how you use the rule. You take your right hand and keep it flat. Point your fingers in the direction of the magnetic field. (Fingers point to the south pole.) Point the thumb in the direction of the velocity of a positively charged particle. When your hand is contorted into this position, your palm will point in the direction of the force that will be acting on the charged particle.

The maximum force, FMax, occurs when the sine of the angle is one (which occurs when = 90),

The right hand rule gives the direction for the force acting on a particle that has a positive charge. If the charge on the particle is negative, then the direction of the force will be in the opposite direction. Or you could use your left hand in the same way.

Time to do a problem or two.

A proton with a velocity of 6.8 x 106 m/s zooms through the earth’s magnetic field. (55 T). What is the max magnetic force acting on the proton?

A proton moving at 5.5 x 107 m/s along the x - axis enters an area where the magnetic field is 3.5 T directed at an angle of 45 to the x - axis lying in the xy plane. (a) What is magnitude of force? (b) What is direction of force? (c) What is the acceleration acting on the proton?

(a) The force is given by:

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(b) Using the right hand rule: Force is in the z direction.

(c) To find the acceleration we use the second law:

Magnetic Force and Work: For work to be done, a force has to act on an object, making it move. It would seem that a magnetic field could do work on a moving charged particle since it does exert a force on it. The other thing to remember, however, is that the force and the displacement have to be in the same direction.

We wrote this as:

The cosine of a ninety degree angle is zero. So for the magnetic force, the work will be zero since the angle between the force and the motion is 90.

A magnetic field does no work on a moving charged particle.

The force is always perpendicular to the magnetic field and the velocity. Therefore the force has no component in the direction of motion. Because of this, the magnetic force does no work

The force can only change the direction of the charge’s motion. It cannot change the kinetic energy of the particle. As the Physics Kahuna has previously stated, magnetic fields can only cause a deflection of the path of a moving charged particle.

Also please to remember that no force would be exerted on a charged particle that was at rest with respect to a magnetic field.

Motion of Charged Particle in Magnetic Field: Time to talk about some common conventions used to draw magnetic fields.

If we want to depict a magnetic field that is perpendicular to the sheet of paper that we have drawn the thing on, we can do this in two ways. If the direction of the magnetic field is out of the paper, then we represent the lines of force as little dots. (sometimes with a circle drawn around them). To show a uniform magnetic field, the dots are shown with equal spacing to the adjacent ones.

To show a magnetic field going into the paper, we represent the lines of force with an “”X”. Think of the “X” as being the tail of the vector. A dot is used to represent the head of the vector, indicating that the lines of force are coming out of the paper.

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What happens when a charged particle moving at a constant velocity enters a uniform magnetic field? Well, outside of the field the particle will travel in a straight line – Newton’s first law, you know, objects in motion stay in motion unless acted upon by an outside force. Once the particle enters the field, an outside force does act on it. The magnetic field will exert a force on the particle.

The magnetic force will change the direction of the particle’s motion. The magnetic force will act on the charged particle all the time and will constantly change its direction as long as it is within the field.

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This constant force acting to change the direction of the particle acts as a centripetal force.

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Can we figure out the radius of the curved path the particle will follow?

Yeah, you betcha! In fact, you must be prepared to derive an equation for the radius of the circular path using the second law, the equation for centripetal acceleration, and the magnetic force. You will have all three of these equations available to you.

Here’s what we do: The particle undergoes a centripetal acceleration. The equation for centripetal acceleration is:

This baby is provided on the test, right?

Using the second law, we can find the magnitude for the centripetal force.

Plug the centripetal acceleration into the second law:

The centripetal force is provided by the magnetic force exerted on the particle. The magnetic force is given by:

sinBF qvB

Here the angle is 90 degrees so the sine is 1. Therefore:

Now we can set the centripetal force and the magnetic force equal to each other.

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Solve this equation for the radius:

A proton moves in a circular orbit of radius 15.0 cm in a uniform magnetic field of 0.500 T. If v is perpendicular to the field, find the speed of the proton.

Note, examining the equation, one finds that the radius of the path is proportional to the momentum of the particle.

The momentum is, of course, mv.

The units get pretty hairy – how did the Physics Kahuna get m/s out of meters, Coulombs, Teslas divided by kilograms? Well, he just did. What is being said here is to not worry about it. Use standard units and they will all work out, like we did with electricity.

If the initial direction of particle’s velocity is not perpendicular to the magnetic field, then there will be an angle between the field and the velocity. The path will end up being a type of spiral called a helix. This would be the general path of a charged particle in a magnetic field. The

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circular path is a special case that occurs only when the direction of the particle’s velocity is perpendicular to the field.

A proton in an electric field will experience a force that is parallel to the lines of force. This is shown in the drawing below on the left. Two plates are oppositely charged. A proton traveling perpendicular to the field enters the area between the plates. The proton is deflected towards the negative plate. You can see the plane that the path the proton takes.

The next drawing, the one on the right, shows the path a proton would take in a magnetic field. The proton’s initial velocity is perpendicular to the magnetic lines of force. The proton follows a circular path and is deflected at a right angle to the field and the velocity. The handsome drawing shows you the path and the plane it lies within.

A particle with an unknown mass and charge moves with a constant speed of v = 2.2 x 106 m/s as it passes undeflected through a pair of parallel plates as shown. The plates are separated by a distance of d = 5.0 x 10-3 m, and a constant potential difference V is maintained between them. A uniform magnetic field of B = 1.20 T directed into the page exists between the plates and to the right of them as shown. After the particle passes into the region to the right of the plates where only the magnetic field exists, it trajectory is circular with radius r = 0.10 m.

(a) What is the sign of the particle’s charge? Explain your answer.(b) On the drawing, indicate the direction of the electric field provided by the plates.(c) Determine the magnitude of the potential difference between the plates.

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(d) Determine the ratio of charge to mass (q/m) of the particle.

(a) If the particle is positive, the magnetic force would be up and particle would curve above the plates. Since it goes the other way, it must have negative charge. Between the plates, the negative particle is deflected downwards. Therefore the electric field must force the negative particle up. The direction of the field is the direction a positive test charge would go so the field must be down. This way the particle will be deflected upward by the electric field of the plates.

(b)

(b) Finding the potential difference between the plates:

The electric field is given by

The magnetic force from the magnetic field is:

The electric force from the plates is:

Set the two forces equal:

plug in for E:

(c) Finding the ratio of charge to mass:

From the circular path of the particle in the magnetic field, we know that:

The centripetal force = the magnetic force in the field

Set these two things equal to each other:

Solving for q/m (the charge to mass ratio):

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Path in an Electric/Magnetic Field: What do we need for a charged particle to move with a constant velocity through a crossed electric and magnetic field (i.e., we have both a magnetic and electric field in the same space)?In an electric field the particle will be deflected along the lines of force. In a magnetic field the particle will be deflected perpendicular to the lines of force. To get the particle to have a constant velocity, the magnetic force needs to cancel out the electric force, so they must be in opposite directions. This means that the magnetic field and the electric field have to be perpendicular to each other.

Earth as a Magnet: The earth has a magnetic field of its own. You can imagine the earth as having an enormous bar magnet stuck down the middle with one end sticking up out of the north pole and the other end sticking out of the south pole.

Please observe the lovely drawing to the right. You can see a sphere representing the earth with a bar magnet stuck through it. You will note that the south pole of the magnet is sticking up where we would normally expect to see the north pole.

This is because the north pole of a compass points to the geographic north. This means that the earth’s uppermost magnetic pole must be the south pole. All of which is very confusing.

We get around this by calling the end of the compass that point north the “north

The compass is a small magnet that is free to rotate. When placed in the earth’s magnetic field the needle lines itself up with the lines of force - it points north.

Lines of flux, as can be seen in the drawing, penetrate the surface of the earth. At the poles the lines of force are almost perpendicular to the earth’s surface. As you move towards the equator the angle gets smaller. This angle is called the dip angle. It varies with your location on the earth.

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Dear Cecil:My wife recently had a fiberglass cast removed and was given a metal brace that is strapped on with Velcro. Walking was painful until she put a couple of magnets against the afflicted area under an Ace bandage. Now she is pursuing the optional roller skate attachment to mount on the brace. She claims the magnets have some kind of magical properties that "cancel out" the pain. Do the magnets actually do something, or is the peroxide seeping through her scalp? --Patrick Colby

Cecil Adams replies:I was all set to write this off as the usual baloney but figured I should riffle through the journals just in case. What do you know--at least one study claims that magnets produced salubrious results. Sure. Give me another hour and I could probably come up with a study proving you can use crystals to cure the common cold. But I figure it won't hurt me to play along with the gag just once.The unexpected results were reported in 1997 by Dr. Carlos Vallbona at Baylor College of Medicine. Fifty patients suffering pain in the aftermath of polio were treated by taping small magnets to the affected parts of their bodies. Twenty-nine patients got real magnets and 21 got fakes. The study was double-blind--neither patients nor staff knew who got the real magnets. The patients rated their pain on a ten-point scale before and after a 45-minute therapy session. The patients with real magnets reported a major decrease in pain (from 9.6 to 4.4 on average), while those with fakes reported much less improvement (from 9.5 to 8.4).

The obvious objections to this study: (1) The investigators had previously reported that magnets relieved their own pain and might have been biased. (2) Double-blind or not, it's pretty easy to tell a real magnet from a fake one, and some patients may have told the doctors what they wanted to hear. (3) We're talking about just one study. Previous research into various types of magnetic therapy came up dry.

The real problem with magnetic therapy--and related issues like whether low-level electromagnetic fields have adverse health effects--is that no one's proposed a plausible physiological explanation for how magnetism does its stuff on the body's cells. (I don't mean all that crap in the ads about "negative and positive ion energy levels"; I mean something you could say in the lab without having everyone roll their eyes.) The chief guru of modern magnetic therapy, Dr. Kyochi Nakagawa of Japan, claims that magnets alleviate "magnetic field deficiency syndrome," said to result from the diminishing strength of the earth's magnetic field, which on the plausibility scale rates just above channeling space aliens. You have to be skeptical on general principles--magnets and related therapies have inspired centuries of quackery.

But I'm tired of always being a party pooper. If you want to tape magnets to yourself, you probably won't do yourself any harm (provided you still see a doctor if you've got a serious complaint) and you'll definitely amuse the other people on the bus. Just don't be surprised if the next study says it was in your head all along.

--CECIL ADAMS

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AP Physics – Magnetism 2 If a moving charge has a force exerted on it when it goes through a magnetic field, shouldn’t the force be even more impressive on a stream of particles, like an electric current flowing through a conductor?

Well, by golly, yes!

Each electron has a force exerted on it by the magnetic field. The force is transferred to the conductor by the collisions the electrons have with the atoms in the conductor.

In the drawing below we have a magnetic field between the two magnets. A current carrying wire passes between the magnet and a force is exerted on it, pushing it up. We can use the right hand rule to figure out the direction of the force. Point your fingers from north to south (direction of the field) and your thumb in the direction of the current. Your palm points up and this is the direction of the force. Just like we did with a single charged particle.

The magnitude of the force exerted on a straight length of wire by the magnetic field is given by the following equation.

Where FB is the magnetic force, I is the current, l is the length of the wire, and is the angle between the wire and the magnetic field.

You will have the use of this equation on the AP Physics Test.

If the magnetic field is perpendicular to the wire (like in the drawing above), then the angle is ninety degrees and the sine of the angle is one. This is when the current will have its maximum value.

Note that if the field and the current are in the same direction, no force is exerted on the conductor.

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A wire has a current of 12.5 A. It goes east to west. The magnetic field goes south to north and is horizontal. The magnetic field strength is 55 T. Find the force acting on a 25cm length of wire.

The magnetic field is perpendicular to the direction of the current, so we have a maximum force.

We can find the direction of the force using the right hand rule:

The direction of the force is down.

Magnetic Field and Conductors: We’ve learned that a magnetic field can affect an electric current – exert a force on the conductor. Do you suppose it is possible that an electric current can have some sort of interface with magnetic fields? Pretty good bet that it would, don’t you think?

One of the really fabulous discoveries of the 19th century was made by Hans Christian Oersted in 1820. This is supposed to be one of those serendipitous accidents. Now Oersted was a respected Danish professor of physics – popular with his students (in this wise not resembling the Physics Kahuna in the least). As part of a classroom demonstration, he brought a magnetic compass near a current carrying wire, he did this, the story goes, to show the class that an electric current would have no effect on a compass. Much to the class’s delight and to the detriment of Oersted’s dignity, the compass needle was deflected. When he reversed the direction of the current, the compass needle swung around and pointed in the opposite direction. This was clear evidence that an electric current developed its own magnetic field.

This discovery lead to some very powerful things that basically changed the world! Oersted reported the phenomenon, and then forgot about it. But other scientists picked up on it.

These drawings represent Oersted's experiment. In the first circuit, no electric current is flowing. The compass needle is pointing north. In the second drawing, the switch is closed and current is flowing. The compass needle is deflected at a right angle to the conduction wire. In the last

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drawing, the battery is rotated so that current flows in the clockwise. The compass needle points in the opposite direction.

The French physicist, Andre-Marie Ampere (1775 - 1836) set up two parallel wires. One of them was free to move sideways, back and forth. When both of the wires carried current in the same direction, they attracted each other. If the current flowed in opposite directions, they repelled each other.

He also found that a current passing through a wire that was bent into the shape of a spring had a stronger magnetic effect. The more turns in the helix, the stronger the field.

Francois Arago (1779 -1853), another French physicist, showed that a copper wire which had current flowing through it acted like a magnet and would attract iron filings.

Johann Salomo Cristoph Schweiger (1779 - 1857) showed that the amount of deflection of the needle in the Oersted experiment was proportional to the strength of the current flowing through the conductor. He thus created the first electric current meter, the galvinometer.

The magnetic field around the current carrying wire can be drawn as a series of concentric circles around the wire as shown in the drawing.

Any current carrying conductor will be surrounded by a magnetic field. When the current first begins to flow, there is no electric field. The electrical energy that flows into the wire, initially, is used to build the magnetic field. Once the magnetic field has been constructed, the current will then flow. When a switch is closed, the current is initially opposed as the magnetic field builds up. Once the field is in place, it does not change, the energy that was needed to build it up is no longer needed, and electric current can flow normally. So with DC, once the field is built up, it doesn’t change and remains constant. If the current varies, the magnetic field will also vary.

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A compass needle is a magnet that can rotate to align itself with a magnetic field. It will be deflected by the field. The first drawing on the left (above) shows the lines of force around a wire that has a current flowing through it. In the second drawing, a set of compasses is placed around a conductor. The magnetic needles align themselves with the earth’s magnetic field and point north. In the final drawing current is flowing through the wire. The magnet of each compass aligns itself with the much stronger magnetic field surrounding the conductor.

Right Hand Rule: Another right hand rule is used to find the direction of the magnetic field around a current carrying conductor.

Right hand rule: Clasp the wire with your right hand. The thumb should point in the direction of the current. Your curled fingers circle the wire in and point in the direction of the magnetic field.

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Magnetic Field and a Current Loop: If a current conducting wire is formed into a loop, the magnetic field is intensified. This is because the magnetic flux lines add together. The field increases with each added loop.

When a conductor is formed into many loops - like a coiled spring - it develops, when current flows through the loops, an intensely strong magnetic field. Such devices are called inductors or coils.

The magnetic field in a conductor can be intensified in several ways.

Increase the number of loops or turns. Increase the current. Construct the turns over a highly permeable material (such as soft iron), called a core.

Permeability is a measure of how attractive a material is to magnetic lines of force. Lines of force are attracted to permeable materials and concentrate in such objects.

When a ferromagnetic core makes up the center of the coil, the magnetic field is even greater. Such devices are called electromagnets. Electromagnets have several advantages over permanent magnets. They can develop very intense magnetic fields - much stronger than permanent magnet fields. Also of great importance -- they also can be switched on and off.

You've probably seen cranes in auto wrecking yards that have large electromagnets at their business end. The electromagnet is lowered onto a junk car, the current is switched on, and the electromagnet picks up the thing. The operator moves the vehicle to where it's to be dumped, the current is switched off, the magnetic field vanishes, and the car falls to its doom.

Another useful application of electromagnetism is the solenoid. This is a coil that has a hollow core (these are often called "air cores"). Adjacent to the coil is a soft iron or steel rod that fits into the hollow core. When the solenoid is energized (the current is switched on), it develops a strong magnetic field and pulls the rod into it. This mechanical action is very useful, it can turn switches on and off and control all sorts of things. Cars, appliances, weapons systems, etc. all make great use of solenoids.

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Strength of Magnetic Field: The magnetic field strength around a straight section of a current carrying conductor is given by this equation:

B is the magnetic field strength, is the permeability of free space, I is the current, and r is the distance to the center of the conductor.

The value for the permeability of free space? Okay, here it is:

A long straight wire has a current of 1.5 A. Find the magnitude of the magnetic field at a point that is 5.0 cm from the wire.

Force Between Parallel Conductors: Ampere found that when two current carrying conductors are in the vicinity of each other, they will exert magnetic forces upon one another.

Each of the conductors creates its own magnetic field. These fields, depending on their direction, will either attract or repel each other.

In the drawing, there are two wires separated by a distance d.

Wire number two sets up a magnetic field, B2. This field is equal to:

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This field exerts a force on wire number one. The force is given by:

The direction of this force is down, towards the other wire. Remember that the other wire will also be attracted by a magnetic force of the same magnitude.

If the current direction in each wire is opposite, the two wires will be repelled.

You won’t have equations for this, instead, you will simply use the equation and the

equation for the force exerted by a wire, to figure it out as above.

A 5.00 cm length of wire has a current of 3.50 A. It is 12.0 cm from a second 5.00 cm length of wire that has a current of 4.95 A in the same direction. Find the force of attraction between the two wires.

The magnetic field around the second wire is:

The force it exerts on the first wire is:

Plug in the equation for the magnetic field:

Force on a Loop: A loop that has a current flowing through it will also be affected by a magnetic field. In the drawing below you can see such a loop that is free to rotate about its center.Each side of the loop is in the magnetic field. The current direction in one of the loop sides is up and the direction on the other side is down. The magnetic field exerts a force on each. The forces are in opposite directions, which causes a torque. The loop will rotate. This is the principle behind how the dc electric motor operates.

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As the loop rotates, the force and the direction of the field stays the same, but the direction of the velocity changes.

The force vectors that act on the loop are shown in the drawing below.

As the loop changes position, the torque will vary between zero and some maximum value. The maximum torque will occur as per the first drawing when the loop is moving perpendicular to the lines of force. The minimum torque, zero torque, occurs when the loop is traveling parallel to the lines of magnetic force. This can be seen in the drawing.

The next drawing, the Physics Kahuna believes that it is directly below this paragraph, shows the torque acting on the loop. The maximum torque occurs when the loop is in position 1 - this is where the motion of the loop is perpendicular to the lines of force, i.e., the angle is 90. As the loop rotates the torque decreases because the force is getting smaller. As decreases the torque must also decrease. When is zero, the torque is zero.

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In drawing 2 (above) the torque has decreased as the angle has decreased.

In drawing 3 (above), the torque has gotten much smaller, the angle is much smaller as well, so the force is small. Finally in drawing 4 no torque is exerted as the magnetic force is zero.

Electric Motors: The simplest dc electric motor is made up of a loop in a magnetic field. Please glance at the drawing of the thing below and use it to follow the Physics Kahuna’s convoluted explanation of how a motor works.

The motor in the drawing has only one loop. In reality, motors have hundred or even thousands of loops. But for simplicity’s sake, we will use an example that has just one loop. The ends of the loop are attached to half rings – these are made of copper or phosphor bronze. This is a contraption called a split ring commutator.

Pushed against the rings are two brushes that are connected to a battery. The brushes are made of solid carbon. Springs push the brushes against the split rings. The rings rotate under the brushes which are stationary. The brushes provide a path for current to flow into and out of the rings.

So how do the motor work? Well, electric current flows from the battery into one of the brushes. The brush is pushing against one of the half rings so electricity can flow from the brush into the ring. This gets it into the loop. It flows around the loop to the other ring where it flows into the brush and then back to the battery. Thus there is a path for the electricity to flow.

The loop is conducting current, so it develops a magnetic field. This field attempts to align itself with the permanent magnetic field. This causes a torque which makes the loop spin. It would normally rotate until the torque was zero, at which point the loop would come to rest as its magnetic field would be aligned with the permanent magnetic field. However, the split rings prevent this from happening. Just as the loop reaches the point where the torque will be zero, the open part of the split ring comes under the brushes and the current stops. The loop keeps rotating because of its inertia. The other split ring rotates under the brush and electricity begins to flow again, except that the direction of the current is opposite to what it was before. This applies a torque which pushes the motor in the same direction it was going. The loop keeps spinning.

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B

F1

F2

B

F1

F2

BB

3 4

no fo rce

In real motors, the loop is a coil wrapped around a ferromagnetic armature (usually soft iron). Most motors have more than one coil. Each of the coils is called a pole and most motors have at least three poles - the good ones have five or seven.

Loud Speakers: Another cool application of the force exerted by a magnetic field on a conductor is the classic loudspeaker. The Physics Kahuna will have shown you several different speaker demonstrations.

Here are the parts of a speaker: a flexible cone – made of paper or thin plastic, a magnet base, and a coil.

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N

N

S

S peaker -exp lod ed

S peaker -a ssem b led

m a gn et

co il

coneto am p lifie r

Here is how the thing works. A small electric signal is sent to the coil from an amplifier (the radio, CD player, whatever). This is a signal that varies with the music, that is, the current increases and decreases with the music. The amount of force exerted on the coil by the magnetic field varies with the strength of the current. When the current increases, the force increases, when the current decreases, the force decreases and so on. The coil sits in a slot cut into the magnet. The force exerted on the coil causes it to move back and forth – with the music. This also vibrates the

cone, which puts the sound into the air.

Permeability: Permeability is a property of a material that has to do with how it changes the flux density in a magnetic field from the permeability value of air.

Some materials (like iron) are very permeable to lines of flux. lines of flux are attracted to the material and pass through it rather than through air. material with low permeability would have little effect on lines of flux, material with a high permeability would dramatically change the flux density of the

magnetic field.

In the drawing above, you can see what happens when a permeable object is placed in the field. The lines of force will concentrate in highly permeable materials.

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N S N Ssof t ir on r ing

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Dear Doctor Science, What is the difference between electromagnetism and animal magnetism? When I kiss my boyfriend, which phenomenon causes our lips to stick together? -- Kathy Cooper from St. Clair Shores, MI

Dr. Science responds:Animal magnetism is a furry, damp form of electromagnetism. Unlike its sleek counterpart, animal magnetism needs to be fed and cared for to be effective. Electromagnets can be made permanent, but animal magnetism is as changeable as animals, which partially explains current divorce rates. What you and your boyfriend have been noticing in your lip adhesion probably has more to do with naturally occurring mouth Velcro than magnetism. If you notice the phenomenon more in the winter months, it may be the same thing as getting your lip stuck to a cold swingset or an outdoor water faucet. In either case, carefully add crazy glue and then pull apart quickly.

AP Physics – Induction Not too many years ago the word “induction” meant “the draft” as in Uncle Sam and the army. Well, fortunately for some - although the Physics Kahuna thinks that mandatory service for your country is a good thing (as Martha Stewart would say) - the draft is gone. Still have to register for it, right? Well, only if you’re a male, women don’t have to register. Don’t you think that in these enlightened times that this is an area that needs to be looked into? Whatever happened to sexual equality in this country?

The discovery of the battery was a significant development in the 18th century. Wet and dry cells provided a continuous source of voltage, and electricity development really took off.

Batteries were not the ultimate answer however (they still aren’t). They are expensive – metals and acids are costly – and they don’t last long. Even today, battery power is much more expensive than the electricity the power company delivers to your house through the power lines.

A really cheap source of electricity would be very useful.

In 1831 two physicists, working independently, found a way to make cheap electricity. Joseph Henry (in the good old US of A) and Michael Faraday (in England) discovered electromagnetic induction.

Here’s a description of Faraday’s initial experiment. Henry did pretty much the same thing. Faraday made a coil around one side of an iron ring and connected the coil to a battery. He placed a switch in this circuit to turn the circuit on and off. A second coil of wire was wrapped around the other half of the ring. He connected the second coil to an electric meter.

When the switch was initially closed, current flowed through the first coil and the meter would spike up, indicating that current was also flowing in the second coil as well. But then the meter needle would then fall back to zero, meaning current was no longer flowing.

Electricity was somehow being induced by the magnetic field of the first coil in the second coil.

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The current was present in the second coil for only a short time. The current was a sort of transient thing that quickly disappeared. As soon as the current built up in the first coil, the current in the other coil went away. Then, when the switch was opened, current was also produced in the second coil, but again, only momentarily.

Faraday didn’t really understand the effect, but Henry did. In fact he explained it to Faraday. (Who, for some reason still gets most of the credit.) Here’s the explanation. When the switch was closed and current began to flow, lines of magnetic force were built up as the magnetic field developed. The lines of flux moved outward as they were created. Henry figured that the moving lines of force cut through the second coil and induced a flow of current. Once the lines of force were in place, they no longer moved, so electricity was no longer induced in the second coil. When the switch was opened, the lines of flux collapsed. They were once again moving, cutting through the second coil and inducing electricity again. It was only while there was motion between the conductor and lines of magnetic flux that electricity was induced.

Induced current can be induced in two separate ways: a conductor can be physically moved through a magnetic field or the conductor can be stationary and the magnetic field can be moved (this is what happened in Faraday's experiment). The production of current depends only on the relative motion between the conductor and the magnetic field. In the drawing below a magnet is dropped through a conductor formed into a coil. As the magnet's lines of flux move through the loops in the

coil, it induces current. The amount of current depends on several factors. One factor is the speed of relative motion. The faster the motion, the greater the current. If you move the magnet very slowly, you won't produce hardly any current at all. If the motion is very rapid, more current is produced. Double the speed and you double the current. Double the magnetic field and you would also double the induced current.

Another factor with a coil is the number of turns in the coil. The more turns, the more voltage. Pushing the magnet through twice as many loops produces twice the voltage. And so on.

Sounds like something for nothing, but that ain't the case. It takes energy to push the magnet through the coil. The more loops, the more energy it takes to push the magnet through them. So you have to put work into the system to induce the electricity.

Electromotive Force, emf: The induced voltage is called the emf. The symbol for emf is . Induction actually creates electromotive force , which really isn’t a force, although they call

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N

S

N

S

it that. We learned about internal resistance in batteries and earlier when we studied current. In the problems we will be doing, internal resistance of the loop (or loops) will usually be negligible, so voltage and emf are essentially the same. Figure . In AP B this is normally the case.

Magnetic Flux: Emf is induced by a change in a quantity called the magnetic flux rather than by a change in the magnetic field. Think of the flux as the strength of a magnetic field moving through an area of space, such as a loop of wire.

For a single loop of wire in a uniform magnetic field the magnetic flux through the loop is given by this equation:

is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and is the angle between B and a normal to the plane of the loop.

The magnetic flux is proportional to the number of lines of force passing through the loop. The more lines the bigger the flux.

The unit for magnetic flux is:

Tm2 or Webers

If the loop is perpendicular to the magnetic field ( would equal zero), then the magnetic flux is simply:

This is the maximum value that the flux can have.

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B

n o rm a l

B

n o rm a l

S id e v iewL oop in fie ld

A rea

= 0= BA= 90= 0

B

S id e v iew o f lo opsin m agne tic fie ld

On the AP Physics Test, you will have the flux equation in this form:

It sort of combines the max value (the part of the thing) with the value for when there is an angle between the lines of force and the loop ( part of the deal).

Induced emf: When a conductor is exposed to a change in magnetic flux, an emf is induced in the circuit. To generate electricity you must have a changing flux. Take a loop of wire and move it through or rotate it in a magnetic field. Another way to think of this is that when the number of line of force in a conductor is constant, no emf is induced. If the number of lines of force is changing, then an emf is induced.

The instantaneous emf in the circuit has this definition:

Instantaneous emf induced in circuit = rate of change of magnetic flux through circuit

The equation for the induced emf is:

The changing flux, means the negatively charged electrons in the wire are exposed to a changing magnetic field. The magnetic field exerts a force on them, causing them to be deflected. Since they are within the wire making up the loop, they end up traveling down the wire. The minus sign in this equation is a reminder that the moving electrons create their own magnetic field. (Moving charge is the basis of electromagnetic fields), and that this field is opposite in direction to the magnetic field causing the flux. This is Lenz’s Law.

The polarity of the induced current depends on Lenz’s Law.

Lenz’s Law The polarity of the induced emf produces a current whose magnetic field opposes the field that induced the current.

In the example to the right, a copper ring is dropped through a magnetic field. Before the ring enters the field, there is no induced current (no changing flux). Once the ring starts to enter the field, it experiences a change of magnetic flux and an emf is induced. This causes a counterclockwise current to flow around the ring.

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N o C u rren t

In d u ced C urren t

N o C u rren t

In d u ced C urren t

N o C u rren t

I

I

Once the entire ring is in the field, however, the magnetic flux stays constant, so once again there is no emf and no current.

When the ring falls out of the field, the magnetic flux is changing again, so an emf is induced and current flows – but in the opposite direction. This is because the area in the field is getting smaller. When the ring entered the field, the area was getting bigger.

Once out of the field, there is no flux (let alone a changing flux) and there is no induced current.

More on Lenz’s Law: The Physics Kahuna will have shown you the classic Lenz’s law demonstrations (unless you’re reading ahead, if that’s the case [and what are the odds] then stand by). One of these involved a falling magnet in a thick walled aluminum pipe. You actually saw two different cylinders dropped down the pipe. The first was an aluminum slug. It fell through the pipe at a rate determined by g. The magnet behaved very differently. As it fell – pulled down by the force of gravity – the lines of magnetic flux around the magnet cut through the aluminum wall of the pipe. This changing flux induced an emf. The current sort of swirled around and around in the pipe walls, which gives them their name eddy currents. The eddy currents build up their own magnetic fields, which oppose the magnetic field of the magnet. This generates an upward force that slows the magnet down and it ends up taking a really long time to fall through the pipe.

To determine the direction of the induced magnetic field, you use the right hand rule as before, but you reverse the direction of current flow in you final answer. Remember you only do this reversal in electromagnetic induction.

A single loop has an induced emf as given in the equation. If we add loops, each extra loop supplies the same amount of emf and we can just add them up. So for a coil on N loops or turns, the emf induced would be:

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What is the Origin of “Crackerjack”?

This word is an Americanism dating back to 1895 which means excellent or superb. Crack-a-jack is a variant. The exact origin of the term is unclear

The name brand for the caramel popcorn product comes from the slang usage. The name was trademarked in 1896. In 1908, Jack Norworth wrote the lyrics to Take Me Out to the Ballgame, cementing the candy's place in American culture with the request to "buy me some peanuts and Crackerjack." (Interestingly, when he wrote the words Norworth had never seen a baseball game and would not see one until 1940. The composer of the tune, Albert Von Tizler, would not see a ballgame until 1928.) The sailor and dog appeared on the box in 1918. By the way, the sailor is named Jack and the dog is Bingo. You check out the official CrackerJack website and learn a lot more at: http://www.crackerjack.com.

http://www.idiomsite.com/crackerjack.htm

Here N is the number of loops.

A rectangular loop is pulled through a uniform magnetic field at a constant velocity v. (See the lovely drawing to the right.) The loop is initially outside the field, it is then pulled into and through the field and ends up on the other side of the magnetic field.

An emf will be induced as the loop enters the field and as it leaves the field since the flux will be changing.

The flux is given by:

The induced emf is given by:

For an emf to be induced, the flux must be changing. This means that the field must change or the area of the loop must change. In the example above, the area is changing as the loop enters the field. Thus an emf will be induced. However, once the loop is completely in the field, the flux does not change, so there will be no induced emf. As the loop leaves the field, the area is again changing – getting smaller, so once again there is a changing flux, so emf is induced again.

The area of the loop is a function of the x position of the thing (since it is moving), which is determined by the velocity.

The area of the loop is:

The velocity is:

Plug in the value for x into the area in the flux equation:

The initial flux is zero, so is simply Blvt.

The emf is given by:

We plug the flux into the emf equation:

This is another equation that you will have available to you for the AP Physics test.:399

vl

d B

x

Note no minus sign for the AP Equation sheet, it got dropped.

We can now plot flux and emf as a function of distance, x.

Here is the plot of the thing.

The flux is initially zero. Once the leading edge of the loop enters the field, the magnetic flux begins to increase. Since we have a changing flux, an emf is induced. The maximum value of the emf is Blv.

Once the loop is entirely within the field, the flux is no longer changing, so the emf falls to zero.

Finally the loop begins to leave the field. At this point the flux changes (the area is getting smaller), so once again an emf is induced. This takes place until the loop is completely out of the field. At this point the emf is zero.

The flux begins at zero and gradually increases to a maximum value of Blvt, once the entire loop is in the field, the flux does not change – it stays at the maximum value. It then begins to drop off as

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d B

x

B lv t

- b lv

b lv

x

x

the loop leaves the field. The emf changes from negative to positive because the change in flux goes from positive to negative.

The direction of the current can be found by using the right hand rule and Lenz’s law. The loop is moving through the field and an emf is induced. The current must be such that it opposes the motion that created the current. This means that the magnetic force has to be in the opposite direction from the applied force and the velocity. We know the direction of the field and the direction of the force, so we can use the right hand rule to find the direction of the current. Point the fingers of the right hand in the direction of the field (into the page in our example). Next point the palm to the left – the direction of the force opposing the motion. The thumb points in the direction of the current flow, which, in this example, is up.

The current direction is up, so at the beginning when the forward edge enters the field, the current will be counterclockwise. Current is induced only in the leading edge. The trailing edge is outside of the field and no emf is induced since it isn’t in a magnetic field and no lines of force pass through it. The top and bottom cut through no lines of force either so no emf is induced in them as well.

Once the loop is within the field, there will be no current induced as the flux is constant. (There is no change in the number of lines of force cutting through the loop.)

When the leading edge leaves the field, the flux will change (because the area is changing – getting smaller). Emf is induced in the trailing edge. The current direction will still be up – nothing has changed so far, right hand rule-wise is concerned. is means that the current changes from counterclockwise to clockwise.

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vB

I

Problem Time:

A loop of wire measures 1.5 cm on each side (it’s a sort of square thing). A uniform magnetic field is applied perpendicularly to the loop, taking 0.080 s to go from 0 to 0.80 T. Find the magnitude of the induced emf in the loop.

The magnetic flux is given by:

We can use this equation because the field is perpendicular to the loop. We then plug this in for the flux in the equation for emf:

A 6.0 cm by 6.0 cm square loop of wire is attached to a cart that is moving at a constant speed of 12 m/s. It travels through a uniform magnetic field of 2.5 T. (a) What is the induced emf after it has traveled 5.0 cm into the field? (b) What is the direction of the current, clockwise or counterclockwise? If the resistance of the loop is 1.0 , what is the current in the loop?

(a) Calculating emf:

(b) Finding the direction of the current. We use the right hand rule for this. Point your fingers in the direction of the magnetic field – out of the page in this case. The palm should point in the direction of the force exerted by the induced current. From Lenz’s law we know that this force must oppose the field that created it. So this is to the left. The thumb points in the direction of the current, which is down.

The current is clockwise in the loop.

(c) Calculating the current. Use Ohm’s law. We know the emf, we assume that the emf is equal to the potential difference V. We also know the resistance of the loop.

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1 0 .0 c m

6 .0 cm

F

B

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AFM

FM

Moving Solid Rail in Magnetic Field: Another interesting type of problem that you will be expected to deal with is the moving solid rail problem.

Two conducting rails lie parallel to each other. One end of the each rail is connected to the other with a load R between them. A conducting bar is placed on top of the rails and is pulled at a constant speed across the top of the rails. The entire system lies within a uniform magnetic field B. The bar /rails system forms a loop that is expanding with time. Therefore the area changes.

As the bar moves along, it experiences a changing magnetic flux. An emf is induced in the loop.

The induced emf is given by this formula (which is provided to you on the AP Physics Test).

is the induced emf, B is the magnetic field, l is the length of the bar, and v is the speed of the bar.

Let’s look at the key forces acting on the system. The rod moves because a force is applied to it, pulling it sideways. It is moving at a constant velocity because the applied force is equal to the magnetic force brought about by the magnetic field acting on a moving conductor. So there are two principle forces. Here’s a picture of them.

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R FA pp

R FA ppFM

I

Now let’s apply this to a problem. A force is applied to a conducting rod so that it slides across a pair of conducting rails. A

uniform magnetic field of 2.50 T is directed into the page. The rails are separated by 18.0 cm. The rod is moving at a constant velocity of 8.50 m/s. The resistance of the system is 1.50 . Find the following: (a) The induced emf in the moving rod, (b) the direction of the current through R, (c) the current through R, (d) The magnitude of the applied force needed to keep the rod moving at constant velocity, (e) The power dissipated by the resistor.

All of this sounds very nasty, but each of the questions is actually quite simple. Let’s do it.

(a) Finding the emf:

(b) We must use the right hand rule to find the direction of the current. The field is into the page, the magnetic force from the induced field must be to the left. so the current through the bar is going up. The current is traveling in a counterclockwise direction. The current is going down through R.

(c)

(d) If v is constant, then sum of forces must be zero. Fapp must equal Fm.

(e)

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FA p p

vFA p p

FMl

x

R

IR

Emf in a Rotating Loop: An emf is induced in a loop rotating in a magnetic field. This is the basis for the electric generator.

Generators were the solution to the expensive electricity problem. The generator takes mechanical motion and converts it into electricity. Falling water could turn them and generate huge amounts of electricity. Steam engines could generate electricity……at last, continuous current in large amounts could be easily and cheaply produced. Today, generator are found everywhere - all conventional cars have generators (or similar devices called alternators), power companies blanket the planet with power lines, aircraft generate their own electricity, you can buy portable generators for when your power goes out, and so on.

Faraday built the first generator. It was a copper disk that was spun with a hand crank. The disk passed through a permanent magnet as it spun and produced a continuous supply of electricity that could do useful work.

Modern generators consist of a rotating loop within a magnetic field. The loop is actually a coil with many hundreds, perhaps thousands, of turns. It is rotated by a prime mover. The prime mover can be falling water, a steam turbine, a cow, the wind, a humanoid, etc.

A simple generator would have a permanent magnet to provide the magnetic field and an armature. Slip rings are provided to make a circuit for the generated electricity to flow through to the load. The slip rings are very similar to the commutator in the DC motor (but no splits).

Because the armature rotates in the magnetic field, the voltage that is induced is not constant. When a conductor is moved parallel with the lines of force, it does not cut through them, there is no change in flux, and no voltage is induced.

If the conductor moves perpendicular to the lines of force, a maximum voltage is induced – you have a maximum change in magnetic flux. The loops that make up the generator's armature are rotating in a stationary magnetic field. Some of the time the loop cuts through lines of force and maximum voltage is induced and sometimes they are traveling parallel to the field and no voltage is generated at all. Also, the loops cut the lines of flux one way and then the other, so the voltage they generate changes polarity.

Generators produce AC current.

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slip r ings

When the loop of the armature is at position A (look at the drawing above), it is essentially traveling parallel to the lines of flux and no emf or current is produced. As it rotates from A to B, it begins to cut lines of flux and the induced emf increases. At B, it is cutting the maximum lines of force and maximum emf is induced (it is traveling perpendicular to the lines of flux). Then the voltage and current drop off as it rotates from B to C. At C no current is induced. Then as it continues to rotate, emf is induced, however, the polarity changes because the loop is cutting through the lines of flux in the opposite direction. It builds up to a maximum value at D, then falls off again to zero at E. Then the cycle repeats itself, etc.

Generators are the opposite of motors. Motors convert electrical energy into mechanical energy; generators turn mechanical energy into electrical energy.

DC can also be produced by a generator through the use of a split ring commutator. The split ring commutator reverses the polarity as the armature rotates, thus keeping the polarity of the induced electricity the same.

Essentially a DC motor and a DC generator are the same device. Turn the rotor and you generate electricity. Run electricity into the rotor, and it turns. So a motor is a generator that is run backwards.

The curve of the current and emf versus time looks like a sine wave. This is because both of those quantities are functions of the sine of the angle between a normal to the loop and the magnetic lines of force in the magnetic field.

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N S

- em f, I

A B C D E

+ em f, I

Tim e

Here (in the picture above) is a loop rotating at a constant angular velocity of . The induced emf will vary sinusoidally with time. An equation for the induced emf can be easily developed. Let’s look at the geometry.

The induced emf in a wire, say the segment of the loop BC, is given by:

The velocity, is the component of the velocity that is perpendicular to the field. This is given by:

We can put these together to get:

is the induced emf, B is the magnetic field, v is the linear velocity of the loop, l is the length of the side of the loop, and is the angle between the lines of force and a normal to the loop.

This is the emf induced in the BC section of the loop, but the same emf is also induced in the AD segment of the loop. So the total emf is given by:

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v l

a2

BC

D

A

v

v

B

The value for v can be found from the angular velocity.

We can plug this into our emf equation:

The angular displacement can be found from the equation for angular velocity.

We can plug this into our emf equation:

The quantity is simply the area of the loop, so:

So, for a loop rotating at a constant angular velocity is a uniform magnetic field, the emf is given by:

One can clearly see that the emf is a function of the sine of the angle and time.

Note that the maximum emf will occur when the value of the sine is one, this gives us:

A generator with only one loop would be a rare thing. Generators always have these massive coil things with bunches of turns. Each turn acts like its own loop, so that the emf for a coil of N loops is given by:

The maximum emf for such a rotating coil would be:

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Dear Cecil:Why don't magnets stick to aluminum? --Les, Los Angeles

Cecil replies:It all has to do with electron shells. In a column of general circulation, however, it's always risky to jump straight into a discussion of electron shells. Better we should edge into this.

First some facts. Fact #1: magnets only stick to other magnets. Fact #2: big magnets are made up of jillions of tiny magnets. Fact #3: so are the metals the magnets stick to, notably iron, nickel, and cobalt, which are called ferromagnetic materials. The difference is that in the big magnets the tiny magnets are organized, i.e., they're all lined up with their north poles in one direction and their south poles in the opposite direction. In an ordinary ferromagnetic material, the tiny magnets are scattered every which way, and their magnetic fields cancel each other out, so no magnetism overall.But suppose we enterprisingly place a ferromagnetic material in a strong magnetic field. Voilà, the formerly scrambled atoms line up parallel with one another. The material as a whole becomes magnetized and sticks to the magnet. Aluminum doesn't contain tiny magnets, so there's nothing to get organized and nothing for the big magnet to stick to.

Certain restless intellects out there may now be wondering: what's with this tiny magnet crap, anyway? That's where the electron shells come in. As you may have guessed by now, the tiny magnets we're talking about are individual atoms. Some atoms, such as those in iron, have individual magnetic fields, while others, such as those in aluminum, do not. It all has to do with the electrons.

Electrons may be thought of as spinning, much as the earth does. They spin one way, they develop a magnetic field with north on top and south on the bottom; they spin the opposite way, they develop a magnetic field with north on the bottom and south on top. For convenience, we call the two directions of spin positive and negative.

Most atoms, such as those in aluminum, have half their electrons spinning in one direction and half in the opposite direction. That means the magnetic fields of the individual electrons cancel each other out. But in the ferromagnetic materials things are different. Take a gander at the third subshell of the M shell of iron, for example. (A shell is an electron's orbit. Electrons are rigidly organized into layers of shells, with so many electrons per shell.) What a wacky sight! We find five electrons with a positive spin and one with a negative spin. This gives the iron atom a pronounced magnetic field. You get those iron atoms lined up, you've got yourself a magnet.

Then we get into a little matter requiring a discussion of quantum mechanics. (What's that, you're sorry you asked? Too late now.) Certain non-ferromagnetic materials, such as chromium and manganese, also have uneven numbers of positive- and negative-spinning electrons in their inner electron shells. Each atom of these substances is magnetic, but the substance as a whole is not. Why? Well, in chromium and manganese, each atom with "up" magnetism is paired with an atom of "down" magnetism, canceling out the magnetism of

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the substance as a whole. In iron, however, all the atomic magnets point in the same direction, so it does (or can) have magnetism overall.

What keeps all the iron atoms pointed in the same direction? It's a quantum mechanical effect known as "exchange interaction." The details of this are still being debated but one plausible interpretation goes like this: Let's say the inner shell or "local" electrons of iron Atom A are spinning in such a way that they have "up" magnetism. The local electrons cause the nearby loose electrons floating around in the metal (the "conduction" electrons) to have opposite or "down" magnetism. The conduction electrons in turn cause the local electrons of neighboring iron Atom B to have "up" magnetism. Result: all the atomic magnets point up and the iron is potentially magnetic.

So why are chromium, manganese, et al different? It turns out manganese and chromium atoms are so close together that the local electrons of Atom A force the local electrons of neighboring Atom B to orient themselves in the opposite direction, without any intervening conduction electrons entering into the picture. Thus each "up" atom is paired with a "down" atom, and the material has no magnetism overall.

That's all pretty clear, right? Well, maybe not. But it's about as clear as stuff like this ever gets.

--CECIL ADAMS

AP Physics – Electromagnetic Wrap UpHere are the glorious equations for this wonderful section.

This is the equation for the magnetic force acting on a moving charged particle in a magnetic field. The angle is the angle between the field B and the velocity v, q represents the charge of the particle. If the velocity is perpendicular to the field, then you have a maximum force and it is simply equal to .

This is the equation for the magnitude of the force exerted on a straight length of wire by a magnetic field. Where FB is the magnetic force, I is the current, l is the length of the wire, and is the angle between the wire and the magnetic field.

This is the magnetic field strength around a straight section of a current carrying conductor. B is the magnetic field strength, is the permeability of free space, I is the current, and r is the distance to the center of the conductor.

The value for the permeability of free space is:

This is the equation for magnetic flux. is the magnetic flux, B is the magnetic field strength, A is the area of the loop, and is the angle between B and a normal to the plane of the loop.

This is the equation for induced emf. is the induced emf, is the change in magnetic flux, and t is the change in time.

This is the equation for the induced emf in a loop of width l that is moving into or out of a magnetic field B at a velocity v.

Here’s what you gots to be able to do:411

Magnetostatics

1. Forces on Moving Charges in Magnetic Field

a. You should understand the force experienced by a charged particle in a magnetic field so you can:

(1) Calculate the magnitude and direction of the force in terms of q, v, and B, and explain why the magnetic force can perform no work.

Okay, this is simply using the old equation. The magnetic force can perform no work because the direction of the magnetic force is always perpendicular to the motion of the particle, so the work is always zero.

(2) Deduce the direction of a magnetic field from information about the forces experienced by charged particles moving through that field.

This is simply the old right hand rule.

(3) State and apply the formula for the radius of the circular path of a charge that moves perpendicular to a uniform magnetic field, and derive this formula from Newton’s Second Law and the magnetic force law.

To do this, you simply derive the formula for the centripetal force and set it equal to the magnetic force. Solve for the radius. The handout is quite happy to show you how to do this.

(4) Describe the most general path possible for a charged particle moving in a uniform magnetic field, and describe the motion of a particle that enters a uniform magnetic field moving with specified initial velocity.

The most general path is a helix spiral kind of deal. If the particle’s velocity is perpendicular to the field, then the particle will follow a circular path.

(5) Describe quantitatively under what conditions particles will move with constant velocity through crossed electric and magnetic fields.

The deal here is that the two forces, the electric and magnetic forces, must somehow add up to zero if the particle is to have a constant velocity. So the forces have to be the same magnitude and be in opposite directions. Since the electric force is in the direction of the electric field and the magnetic force is perpendicular to the magnetic field, then the two fields have to be perpendicular to each other.

2. Forces on current-carrying Wires in Magnetic Fields

a. You should understand the force experienced by a current in a magnetic field so you can:

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(1) Calculate the magnitude and direction of the force on a straight segment of current-carrying wire in a uniformed magnetic field

Just use the equation;

(2) Indicate the direction of magnetic forces on a current-carrying loop of wire in a magnetic field, and determine how the loop will tend to rotate as a consequence of these forces.

Use the good old right hand rule.

3. Fields of Long Current-carrying Wires

a. You should understand the magnitude field produced by a long straight current-carrying wire so you can:

(1) Calculate the magnitude and direction of the field at a point in the vicinity of such a wire.

Use the equation.

(2) Use superposition to determine the magnetic field produced by two long wires.

Superposition means vector addition. Each of the wires produces its own field. You just add

them up as vectors.

(3) Calculate the force of attraction or repulsion between two long current carrying wires.

Use the equation and the equation for the force exerted by a wire,

.

Electromagnetism

1. Electromagnetic Induction

a. You should understand the concept of magnetic flux so you can:

(1) Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation.

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Simply use the equation for magnetic flux.

b. You should understand Faraday’s Law and Lenz’s Law so you can:

(1) Recognize situations in which changing flux through a loop will cause an induced emf or current in the loop.

Faraday’s law is simply that a current is induced when the magnetic flux through the conductor is changing. Have to be alert for the idea that it is only when the flux is changing that an emf will be induced. Typical situations where this happens would be where a loop travels into or out of a magnetic field. Please note that once it is in a field, even though it is moving, no emf is induced because the flux is not changing. The other situation is a loop that is rotating within a magnetic field.

(2) Calculate the magnitude and direction of the induced emf and current in:

(a) A square loop of wire pulled at a constant velocity into or out of a uniform magnetic field.

Use the equation . To figure out the current direction, use the right hand rule.

(b) A loop of wire placed in a spatially uniform magnetic field whose magnitude is changing at a constant rate.

Use the equation . (c) A loop of wire that rotates at a constant rate about an axis perpendicular to a

uniform magnetic field.

This is a bit more complicated. There are some examples of how to do this in the last

electromagnetic handout. Check it out.

(d) A conducting bar moving perpendicular to a uniform magnetic field.

Just use the good old equation.

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From 2000:

7. A particle with unknown mass and charge moves with constant speed V = 1.9 x 106 m/s as it passes undeflected through a pair of parallel plates, as shown above. The plates are separated by a distance d = 6.0 x 10-3 m, and a constant potential difference V is maintained between them. A uniform magnetic field of magnitude B = 0.20 T directed into the page exists both between the plates and in a region to the right of them as shown. After the particle passes into the region to the right of the plates where only the magnetic field exists, its trajectory is circular with radius r = 0.10 m.

(a) What is the sign of the charge of the particle? Check the appropriate space below.

___Positive _xx_ Negative ___ Neutral ___It cannot be determined from this information.

If the particle was positive, the magnetic force would be up (right hand rule) and the particle would curve above the plates, it’s motion is the opposite, so the particle must have a negative charge. Between the plates, the negative particle is deflected downward. Therefore the electric field must force the negative particle up, but direction of the field is the direction a positive test charge would go so the field must be down.

(b) On the diagram above, clearly indicate the direction of the electric field between the plates.

(c) Determine the magnitude of the potential difference V between the plates.

The magnetic field strength is given by:

Magnetic force:

Electric Force:

Set the electric force and the magnetic force equal to one another:

(d) Determine the ratio of the charge to the mass ( q/m) of the particle.

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v

VA

R

This sounds bad. Okay, the particle follows a curved path in the magnetic field. So we can set the centripetal force = magnetic force.

Solving for q/m

From 1999:

A rectangular conducting loop of width w, height h, and resistance R is mounted vertically on a nonconducting cart as shown above. The cart is placed on the inclined portion of a track and released from rest at position P1 at a height yo above the horizontal portion of the track. It rolls with negligible friction down the incline and through a uniform magnetic field B in the region above the horizontal portion of the track. The conducting loop is in the plane of the page, and the magnetic field is directed into the page. The loop passes completely through the field with a negligible change in speed. Express your answers in terms of the given quantities and fundamental constants.

(a) Determine the speed of the cart when it reaches the horizontal portion of the track.

(b) Determine the following for the time at which the cart is at position P2 with one-third of the loop in the magnetic field.

i. The magnitude of the emf induced in the conducting loop

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ii. The magnitude of the current induced in the conducting loop

(c) On the following diagram of the conducting loop, indicate the direction of the current when it is at position P2.

(d) i. Using the axes below, sketch a graph of the magnitude of the magnetic flux through the loop as a function of the horizontal distance x traveled by the cart, letting x = 0 be the position at which the front edge of the loop just enters the field. Label appropriate values on the vertical axis.

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B w h

2 oB h gyR

2 oB h gyR

ii. Using the axes below, sketch a graph of the current induced in the loop as a function of the horizontal distance x traveled by the cart, letting x = 0 be the position at which the front edge of the loop just enters the field. Let counterclockwise current be positive and label appropriate values on the vertical axis.

From 1996:

A rigid rod of mass m and length is suspended from

two identical springs of negligible mass as shown in the diagram below. The upper ends of the springs are fixed in place and the springs stretch a distance d under the weight of the suspended rod.

(a) Determine the spring constant k of each spring in terms of the other given quantities and fundamental constants.

But each spring holds ½ the total mass

As shown below, the upper end of the springs are connected by a circuit branch containing a battery of emf and a switch S, so that a complete circuit is formed with the metal rod and springs. The circuit has a total resistance R, represented by the resistor in the diagram below. The rod is in a uniform magnetic field, directed perpendicular to the page. The upper ends of the springs remain fixed in place and the switch S is closed. When the system comes to equilibrium, the rod is lowered an additional distance d.

(b) What is the direction of the magnetic field relative to the coordinate axes shown on the right in the previous diagram?

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Using the right hand rule, the field is out of the page, along the positive z axis. The current is to the right – thumb points that way – and the bar is pushed down, so the palm points down. The fingers end up pointing out of the page.

(c) Determine the magnitude of the magnetic field in terms of m, , d, d, , R, and fundamental constants.

Gravity initially displaced the bar a distance d. Then the magnetic field displaced the bar an extra distance, d. Set the spring force equal to the magnetic force:

But, there are two springs

x is d, so plug that in: Solve for B:

Now we need to get B in terms of the items listed in (c):

If the internal resistance is not mentioned, we can assume that it is zero. So

Plug this into the equation we developed for I:

Also, we already found a value for

Plug this into the equation for k:

(d) When the switch is suddenly opened, the rod oscillates. For these oscillations, determine the following quantities in terms of d, d, and fundamental constants:

i. The period.

But there are two springs.

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Plug in the value for k.

ii. The maximum speed of the rod.

Use conservation of energy:

The 2 in front of US because there are two springs

and

From 1995:

In a linear accelerator, protons are accelerated from rest through a potential difference to a speed of approximately 3.1 x 106 meters per second. The resulting proton beam produces a current of 2.0 x 10-6ampere.

a. Determine the potential difference through which the protons were accelerated.

The electric potential energy is equal to the work done on the particle. This must equal its kinetic energy.

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Note, we can look up the mass of a proton (it is 1.67 x 10-27 kg).

b. If the beam is stopped in a target, determine the amount of thermal energy that is produced in the target in one minute.

Because energy is conserved, the thermal energy produced in the target is simply equal to the power of the beam of particles.

The proton beam enters a region of uniform magnetic field B, as shown above, that causes the beam to follow a semicircular path.

c. Determine the magnitude of the field that is required to cause an arc of radius 0.10 meter.

d. What is the direction of the magnetic field relative to the axes shown above on the right?

Positive Z -- out of the paper..

Use the right hand rule. Thumb points in direction of velocity, palm points in direction of magnetic force, and the fingers point in the direction of the field.

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At the dot on the curved path of the proton, the palm points to the left – towards the center of the circle. The thumb points down (the velocity at that point is a tangent to the circle and is down). Your fingers end up pointing out of the page.

From 1994: A force F is applied to a conducting rod so that the rod slides with constant speed v over a

frictionless pair of parallel conducting rails that are separated by a distance . The rod and rails have negligible resistance, but the rails are connected by a resistance R, as shown below. There is a uniform magnetic field B perpendicular to and directed out of the plane of the paper.

a. On the following diagram, indicate the direction of the induced current in the resistor.

Using the right hand rule, the palm must point to the left (the direction of the force which opposes the motion of the bar), the fingers point in the direction of the field – out of the page. This thumb then points in the direction of the current through the bar, which is down. Therefore the current is clockwise and the current through the resistor is up.

Determine expressions for the following in terms of v, B, , and R. b. The induced emf in the rod.

c. The electric field in the rod.

d. The magnitude of the induced current in the resistor R.

We know that the emf induced is:

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We also know that, if the resistance in the rod and rails is negligible, then the emf is equal to the voltage.

So:

Now we can apply Ohm’s law.

e. The power dissipated in the resistor as the rod moves in the magnetic field.

f. The magnitude of the external force F applied to the rod to keep it moving with constant speed v.

At constant speed the net force must equal zero.

From 1998:

The long, straight wire shown in Figure 1 below is in the plane of the page and carries a current I. Point P is also in the plane of the page and is a perpendicular distance d from the wire. Gravitational effects are negligible.

a. With reference to the coordinate system in Figure 1, what is the direction of the magnetic

field at point P due to the current in the wire?

Use the right hand rule. Into the page, in the minus z direction.

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A particle of mass m and positive charge q is initially moving parallel to the wire with a speed vo when it is at point P, as shown in Figure 2 below.

b. With reference to the coordinate system in Figure 2, what is the direction of the magnetic force acting on the particle at point P?

Right hand rule. Force is down, in the minus y direction. Thumb points in direction of v0 , fingers point into page (direction of field), so the palm points down, the direction of the force.

c. Determine the magnitude of the magnetic force acting on the particle at point P in terms of the given quantities and fundamental constants.

d. An electric field is applied that causes the net force on the particle to be zero at point P.i. With reference to the coordinate system in Figure 2, what is the direction of the

electric field at point P that could accomplish this?

The electric field direction is defined as the direction a positive particle moves. The positive particle must move up to counter the downward magnetic field.

ii. Determine the magnitude of the electric field in terms of the given quantities and fundamental constants.

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From 1992:

The figure shows a cross section of a cathode ray tube. An electron in the tube initially moves horizontally in the plane of the cross section at a speed of 2.0 x 107 meters per second. The electron is deflected upward by a magnetic field that has a field strength of 6.0 x 10-4 Tesla.

a. What is the direction of the magnetic field?

Out of the page.

b. Determine the magnitude of the magnetic force acting on the electron.

c. Determine the radius of curvature of the path followed by the electron while it is in the magnetic field.

An electric field is later established in the same region as the magnetic field such that the electron now passes through the magnetic and electric fields without deflection.

d. Determine the magnitude of the electric field.

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e. What is the direction of the electric field?

The force needs to be downward on the electron to cancel the upward magnetic field force. Unlike the magnetic field which is perpendicular, the electric field is parallel to its force. The electric field is the direction a positive charge moves. Electrons move opposite the field. So if the electron needs to move down the electric field is up.

From 1993:

A particle of mass m and charge q is accelerated from rest in the plane of the page through a potential difference V between two parallel plates as shown. The particle is injected through a hole in the right-hand plate into a region of space containing a uniform magnetic field of magnitude B oriented perpendicular to the plane of the page. The particle curves in a semicircular path and strikes a detector. Neglect relativistic effects throughout this problem.

a. Determine the following

i. State whether the sign of the charge on the particle is positive or negative.

Positive. The right plate is negative and the left is positive. The particle is attracted to the negative plate and repelled by the positive plate so it has to be positive.

ii. State whether the direction of the magnetic field is into the page or out of the page.

Out of the page. Look at the particle at the mid point of the curve (where the little arrowhead is located). Its velocity is down, so point your thumb down. It is being forced towards the center by the magnetic force, so point your palm to the left. This leaves your fingers pointing out of the page, which is the direction of the magnetic field.

b. Determine each of the following in terms of m, q, V, and B.

i. The speed of the charged particle as it enters the region of the magnetic field B.

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ii. The force exerted on the charged particle by the magnetic field B.

iii. The distance from the point of injection to the detector.

iv. The work done by the magnetic field on the charged particle during the semicircular trip.

No work is done since the magnetic field is perpendicular to the displacement.

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