Magnetic Fields. Definition : A magnetic field is a force field which surrounds either a magnet or a wire carrying an electric current and will act upon,

Magnetic Fields

Magnetic Fields

Definition : A magnetic field is a force field which surrounds either a magnet or a wire carrying an electric current and will act upon, without contact, another magnet or current carrying wire

Like the other fields we have studied we represent magnetic fields diagrammatically using field lines or lines of magnetic flux

Magnetic FieldsWe name the ends of a magnet “the poles”. (North and & South). More correctly they should be referred to as the “North seeking pole” and “South seeking pole”

The arrows on a magnetic field line represent the path which a “small, free north pole” would take

Like poles repel each other

Unlike poles attract each

other

The Earth’s Magnetic Field

The Earth has a magnetic field just like a giant magnet.

The geographic North pole has a South magnetic pole associated with it (Since a north seeking pole of a magnet will point towards it)

The Earth’s Magnetic Field

Watch terminology!!!

At the geographic North pole there is a magnetic pole which we can refer to as “Magnetic North”. However it is a South pole!

Watch the field lines!

Geographic North Pole

Geographic South Pole

Magnetic Fields around Wires

Wires carrying an electric current produce a magnetic field.

The “right-hand rule” can be used to establish the direction of this field. Note the current direction is the direction of “conventional current” positive to negative

The Motor EffectA current carrying wire, with its associated magnetic field will experience a “motor effect” if placed (at a non-zero angle) in a magnetic field.

The force is perpendicular to both the current & the magnetic field

Factors Affecting the Motor Effect

Experimentally, it can be shown that the size of the force due to the motor effect is related to the following :

1. The strength of the current

2. The strength of the magnetic field

3. The length of the wire

4. The angle between the field lines & current

In terms of angles, the force is greatest when the current is perpendicular to the magnetic field and zero when parallel to the field

Fleming’s Left-Hand RuleThe relationship between field, current and force can best be remembered using “Fleming's left hand rule”

First finger (Field), seCond Finger (Current), Thumb (moTion)

moTion

Magnetic Flux Density

For a given magnetic field, the Force on a current-carrying wire (when perpendicular to the field) is equal to:

Where is the current, is the length of the wire, and B is the Magnetic flux density, equal to the force per unit length per unit current.

Unit of Magnetic flux: NA-1m-1 but given the unit of Tesla (T) (therefore 1T = 1 NA-1m-1)

AnswersForce Magnetic flux

densityCurrent Length

20 N 0.3 T 7 A 9.52 m

3 N 1 mT 20 A 150 m

500 N 33 T 200 mA 75.8 m

67 N 0.4 T 2.5 A 67 m

6 kN 4 T 60 A 25 m

Complete:

Question 1A straight horizontal wire of length 5m is in a uniform magnetic field which has a magnetic flux density of 120mT. The wire is perpendicular to the field lines which act due North. When the wire conducts a current of 14A from East to West calculate the magnitude and direction of the force on the wire.

F = 120 x 10-3 x 14 x 5

F = 8.4N

Vertically downwards

Current East to West

Field due North

Problems

2.4 x 10-2N West

4.5A East to West

0.2T Vertically

downSouth

8.0 x 10-3N

Electric MotorsThe force that a current carrying conductor experiences in a magnetic field is the basic principle behind an electric motor.

N S

Consider the above rectangular coil which has n turns & can rotate about its vertical axis. The coil is arranged in a uniform magnetic field. The coil will experience a pair of forces where the direction is given by the left hand rule.

Force(out of page)

Force(into page)

X Y

Electric MotorsThe vertical sides of the coil are perpendicular to the field and will experience a force given by:

Each side will experience a force in the opposite direction. Since we have a coil with turns the force is given by:

We have a pair of forces in opposite directions (i.e. we have a Torque equal to Fd where d is the perpendicular distance separating the forces. In this case d is the width of the coil w)

Electric MotorsLooking from above :

F

F

w

Coil Parallel to field. Torque =

F

F

Coil now at an angle to the field. Torque =

But

is the area of the coil giving

When the coil is parallel to field ( =0, cos 0 = 1), the torque is BIAn. When the coil is perpendicular ( =90, cos 90 = 0), the torque is zero.

X

Y

X

Y

Electric Motors: Practicalities

In real motors, current must be delivered to the rotating coil (direct connections would twist!). In simple motors, sprung loaded carbon brushes push against a rotating commutator

The second issue is that after half a rotation the force would change direction. We need to change the direction of the current so that as the coil rotates the force is always in the same direction. A split-ring commutator is used

Question 2A rectangular coil with 50 turns of width 60mm & length 80mm is placed parallel to a uniform magnetic field which has a flux density of 85mT. The coil carries a current of 8A and the shorter side of the coil is parallel to the field.Sketch the arrangement and determine the Torque acting on the coil

0.085 x 8 x 0.06 x 0.08 x 50 x cos(0)

(0.2Nm)

Part 7b- Charged Particles in Magnetic Fields

Moving Charges in Magnetic FieldsCurrent carrying conductors experience a force in a magnetic field. In a similar way charged particles also experience a force in a magnetic field and are deflected

This technology has been exploited in all sorts of Cathode Ray Tubes (CRTs), TVs, Monitors & Oscilloscopes

Moving Charges in Magnetic Fields

The subtopics are a bit back to front... The reason why a current carrying conductor experiences a force is because the electrons moving along the wire experience a force and are moved to one side of the conductor, which exerts a force on it

- - - - - - - -F F F F F F F F

Moving Charges in Magnetic FieldsA beam of charged particles is a flow of electric current (current = ).

Consider a charge moving with a velocity in a time . The distance travelled is therefore .

The above equation defines the force experienced by a particle with a charge of Q as it moves with a velocity v in a perpendicular direction to a magnetic field with flux density B

Question 1Electrons move upwards in a vertical wire at ms-1 into a uniform horizontal magnetic field which has a flux density of 95 mT & is oriented along a line South to North.Calculate the magnitude and direction of the force on each electron.

Due East

Path of a Charged Particle in a FieldIn the previous lesson we’ve seen that a moving charged particle is deflected in a magnetic field, in accordance with Fleming’s left hand rule

Electron Gun

Magnetic field coming out of

the page

Force

ConventionalCurrent

Electron Gun

Force

ConventionalCurrent

Electron Gun

Force

The force always acts perpendicular to the velocity, causing the path to change...

Circular motion is achieved!

ConventionalCurrent

Path of a Charged Particle in a FieldAs always with circular motion problems we are looking for the force to “equate” to the centripetal force

From earlier, we know the force on a charged particle: F= BQv

BQv = mv2 / r

r = mv/BQ

The path therefore becomes more curved (r reduced) if the flux density increases, the velocity is decreased, or if particles with a larger specific charge (Q/m) are used

Instantaneous Velocity

BQv

Question 2A beam of electrons with a velocity of m/s is fired into a uniform magnetic field which has a flux density of 8.5mT. The initial velocity is perpendicular to the field. (a) Calculate the radius of the orbit.(b) What must the flux density be adjusted to if the

radius of the orbit is desired to be 65mm?(a)

(b)

Uses of Magnetic FieldsYour task is to work in groups of 2-3, or individually, to investigate one application or use of electromagnets. You will have all lessons this week, plus prep, to prepare a presentation, which will be given on Tuesday 18th March.

If you don’t know where to start, there are some ideas in the book! Feel free to be adventurous though… And try to get some equations in there!

- Thermionic emission - Superconducting magnets

- Maglev - Synchrotrons

- Mass spectrometers - Particle accelerators

Part 7c- Applications of Magnetic Fields

Application 1 : CRTLast lesson we mentioned the CRT (Cathode Ray Tube)

Such devices can also be called:

- Electron guns- Thermionic devices

The cathode is a heated filament with a negative potential which emits electrons, a nearby positive anode attracts these electrons which pass through a hole in the anode to form a beam. This is called Thermionic emission. The potential difference between the anode and cathode controls the speed of the electrons.

Applications : Mass SpectrometersA very clever component known as a “velocity selector” is used to obtain a constant velocity.

Firstly, we ionise the atoms.

The +ve ions are acted upon by both an electric field & a magnetic field.

The Electric Force (upwards) is given by & the Magnetic Force (downwards) is given by . Only particles that have will make it through the slit (and therefore only particles that have or ).

Applications : Mass Spectrometers

Applications : Mass Spectrometers

These machines can then be used to analyse the types of atoms, (and isotopes) present in a sample, using:

𝑟=𝑚𝑣𝐵𝑄

Different ions have different specific charges () and are therefore deflected at different radii (). We can then measure the relative abundance of each ion.

Applications : CyclotronsCyclotrons are a method of producing high energy beams used for nuclear physics & radiation therapy

An alternating electric field is used to accelerate the particles while a magnetic field causes the particles to move in a circle, (forming a spiral)

Compared to a linear accelerator, this arrangement allows a greater amount of acceleration in a more compact space

Applications : Cyclotrons

Two hollow D shaped electrodes exist in a vacuum. A uniform magnetic field is applied perpendicular to the plane of the “D”s

Charged particles are injected into a D, the magnetic field sets the particle on a circular path causing it to emerge from the other side of this D & to enter the next.

As the particle crosses the gap between the Ds the supplied current changes direction (high frequency AC) & the particle is accelerated, (causing a larger radius each time)

Applications : CyclotronsAssuming Newtonian rather than relativistic velocities apply…The particle leaves the cyclotron when the velocity causes the path radius to equal the radius R of the D:

The period for one cycle of the AC must be approximate to the time for one complete circle (2R). Using

The frequency of the AC will be

Charged Particles in Magnetic Fields

Moving Charges in Magnetic FieldsCurrent carrying conductors experience a force in a magnetic field. In a similar way charged particles also experience a force in a magnetic field and are deflected

This technology has been exploited in all sorts of Cathode Ray Tubes (CRTs), TVs, Monitors & Oscilloscopes

Moving Charges in Magnetic Fields

The subtopics are a bit back to front... The reason why a current carrying conductor experiences a force is because the electrons moving along the wire experience a force and are moved to one side of the conductor, which exerts a force on it

- - - - - - - -F F F F F F F F

Moving Charges in Magnetic FieldsA beam of charged particles is a flow of electric current (current = ).

Consider a charge moving with a velocity in a time . The distance travelled is therefore .

The above equation defines the force experienced by a particle with a charge of Q as it moves with a velocity v in a perpendicular direction to a magnetic field with flux density B

Question 1Electrons move upwards in a vertical wire at ms-1 into a uniform horizontal magnetic field which has a flux density of 95 mT & is oriented along a line South to North.Calculate the magnitude and direction of the force on each electron.

Due East

Path of a Charged Particle in a FieldIn the previous lesson we’ve seen that a moving charged particle is deflected in a magnetic field, in accordance with Fleming’s left hand rule

Electron Gun

Magnetic field coming out of

the page

Force

ConventionalCurrent

Electron Gun

Force

ConventionalCurrent

Electron Gun

Force

The force always acts perpendicular to the velocity, causing the path to change...

Circular motion is achieved!

ConventionalCurrent

Path of a Charged Particle in a FieldAs always with circular motion problems we are looking for the force to “equate” to the centripetal force

From earlier, we know the force on a charged particle: F= BQv

BQv = mv2 / r

r = mv/BQ

The path therefore becomes more curved (r reduced) if the flux density increases, the velocity is decreased, or if particles with a larger specific charge (Q/m) are used

Instantaneous Velocity

BQv

Question 2A beam of electrons with a velocity of m/s is fired into a uniform magnetic field which has a flux density of 8.5mT. The initial velocity is perpendicular to the field. (a) Calculate the radius of the orbit.(b) What must the flux density be adjusted to if the

radius of the orbit is desired to be 65mm?(a)

(b)

Applications of Magnetic Fields

Application 1 : CRTLast lesson we mentioned the CRT (Cathode Ray Tube)

Such devices can also be called:

- Electron guns- Thermionic devices

The cathode is a heated filament with a negative potential which emits electrons, a nearby positive anode attracts these electrons which pass through a hole in the anode to form a beam. This is called Thermionic emission. The potential difference between the anode and cathode controls the speed of the electrons.

Applications : Mass SpectrometersA very clever component known as a “velocity selector” is used to obtain a constant velocity.

Firstly, we ionise the atoms.

The +ve ions are acted upon by both an electric field & a magnetic field.

The Electric Force (upwards) is given by & the Magnetic Force (downwards) is given by . Only particles that have will make it through the slit (and therefore only particles that have or ).

Applications : Mass Spectrometers

Applications : Mass Spectrometers

These machines can then be used to analyse the types of atoms, (and isotopes) present in a sample, using:

𝑟=𝑚𝑣𝐵𝑄

Different ions have different specific charges () and are therefore deflected at different radii (). We can then measure the relative abundance of each ion.

Applications : CyclotronsCyclotrons are a method of producing high energy beams used for nuclear physics & radiation therapy

An alternating electric field is used to accelerate the particles while a magnetic field causes the particles to move in a circle, (forming a spiral)

Compared to a linear accelerator, this arrangement allows a greater amount of acceleration in a more compact space

Applications : Cyclotrons

Two hollow D shaped electrodes exist in a vacuum. A uniform magnetic field is applied perpendicular to the plane of the “D”s

Charged particles are injected into a D, the magnetic field sets the particle on a circular path causing it to emerge from the other side of this D & to enter the next.

As the particle crosses the gap between the Ds the supplied current changes direction (high frequency AC) & the particle is accelerated, (causing a larger radius each time)

Applications : CyclotronsAssuming Newtonian rather than relativistic velocities apply…The particle leaves the cyclotron when the velocity causes the path radius to equal the radius R of the D:

The period for one cycle of the AC must be approximate to the time for one complete circle (2R). Using

The frequency of the AC will be