Topic 6.2 Extended B – Electric field

Topic 6.2 ExtendedB – Electric field

The normal force, the friction force, tension and drag are all classified as contact forces, occurring where two objects contact each other.The gravitational force, and the electric force, however, do not need objects to be in contact (or even close proximity). These two forces are sometimes called action at a distance forces.There are two schools of thought on action at a distance.

School A: The masses know where each other are at all times, and the force is instantaneously felt by both masses at all times.School B: The masses deform space itself, and the force is simply a reaction to the local space, rather than the distant mass.

FYI: School A must assume that the force "signal" travels infinitely fast, to react to relative motion between the masses.

FYI: School B assumes that the force "signal" is already in place in the space surrounding the source of the force. Thus the force signal does NOT have to keep traveling to the receiving mass.

FYI: Relativity has determined that the absolute fastest ANY signal can propogate is c = 3.0108 m/s, the speed of light.

Thus School B is currently the "correct" view. If, for example, the gravitational forces were truly "action at a distance," the following would occur:

Topic 6.2 ExtendedB – Electric field

SUN c

c

The planet and the sun could not "communicate" quickly enough to keep the planet in a stable orbit.

Topic 6.2 ExtendedB – Electric field

Instead, look at the School B view of the space surrounding the sun:

The planet "knows" which way to "roll" because of the local curvature of the space surrounding the sun.

Long distance communication is not needed.

FYI: You have probably seen such a model at the museum. A coin is allowed to roll, and it appears to be in orbit.

FYI: Of course, if the slope isn't just right to match the tangential speed, the coin will spiral into the central maw.

FYI: The view of School B is called the FIELD VIEW.

Of course, if the mass isn't moving, it will roll downhill if placed on the grid:

Topic 6.2 ExtendedB – Electric field

Depending on where we place the "test" mass m0, it will roll differently.

m0

m0

Question: Why is the arrow at the location of the first test mass smaller than that at the second one?

Topic 6.2 ExtendedB – Electric field

We represent fields with vectors of scale length.

In the case of the gravitational field, the field vectors all point toward the center:

We can assign an arrow to each position surrounding the sun, representing the direction the test mass will go, and how big a force the test mass feels.

If we view our gravitational field arrows from above, we get a picture that looks like this:

Topic 6.2 ExtendedB – Electric field

SUN

FYI: We don't even have to draw the object that is creating the field.

What the field arrows tell us is the magnitude and the direction of the force on a particle placed anywhere in space:

The blue particle will feel a "downward" force.

The red particle will feel a "leftward" force whose magnitude is LESS than that of the blue particle.

Topic 6.2 ExtendedB – Electric field

For masses, all we have is an attractive force.

SUN

All of the field arrows point inward.

Consider, now, a negative charge:

-If we place a very small POSITIVE test charge in the vicinity of our negative charge, it will be attracted to the center: We can map out the field vectors just as we did for the sun.

FYI: Inward-pointing fields are called SINKS. Think of the field arrows as water flowing into a hole.

FYI: Test charges are by convention POSITIVE. Therefore, field vectors around a charge show the direction a POSITIVE charge would want to go if placed in the field.

Topic 6.2 ExtendedB – Electric field

Now suppose our charge is positive.

+If we place a very small POSITIVE test charge in the vicinity of our positive charge, it will be repelled from the center: We can map out the field vectors just as we did for the negative charge.

FYI: Outward-pointing fields are called SOURCES. Think of the field arrows as water flowing out of a fountain.

Question: T or F: Gravitational fields are always sinks.

Question: T or F: Electric fields are always sinks.We can place a negative charge here.

-The fields of the two charges will interact: Keep in mind that the field lines show the direction a POSITIVE charge would like to travel if placed in the field.

FYI: We call such a system an ELECTRIC DIPOLE. A dipole has two poles (charges) which are opposite in sign.

FYI: The ELECTRIC DIPOLE consists of a source and a sink.

+

-

A negative charge will do the opposite:

You may have noticed that in the last field diagram some of our arrows were unbroken.

Topic 6.2 ExtendedB – Electric field

+We can draw solid field arrows as long as we note that THE CLOSER THE FIELD LINES ARE TO ONE ANOTHER, THE STRONGER THE FIELD.

-

strong

weak

FYI: There are some simple rules for drawing these solid "electric lines of force."

Rule 1: The closer together the electric lines of force are, the stronger the electric field.

Rule 2: Electric lines of force originate on positive charges and end on negative charges.

Rule 3: The number of lines of force entering or leaving a charge is proportional to the magnitude of that charge.

Topic 6.2 ExtendedB – Electric field

Determine the sign and the magnitude of the charges by looking at the electric lines of force.

C

D

A B

E F

Question: Which charge is the strongest negative one? D

Question: Which charge is the weakest positive one? E

Question: Which charge is the strongest positive one? A

Question: Which charge is the weakest negative one? C

So how do we define the magnitude of the electric field vector E?

Topic 6.2 ExtendedB – Electric field

Here's how:

E = F on qo

q0 where q0 is the positive test charge

Electric field definition

Think of the electric field as the force per unit charge. To obtain a more useful form for E, consider the force F between a test charge q0 and an arbitrary charge q: F = kq0q

r2

then E = F on qo

q0

= kq0q

q0r2=

kqr2

so that

E = kqr2

Electric field in space surrounding a point charge

Topic 6.2 ExtendedB – Electric field

What is the magnitude of the electric field strength two meters from a +100 C charge?Use

E = kqr2

=

9109·10010-6 22

= 225000 n/C

What is the force acting on a +5 C charge placed at this position?

From E = Fon qo

q0

we see that F = qE Force on a charge placed in an electric field

so that F = qE = 510-6·225000

= 1.125 n

Topic 6.2 ExtendedB – Electric field

What is the electric field at the chargeless corner of the 2-meter by 2-meter square?

Start by labeling the charges (organize your effort):

|EA| = kqr2

=

9109·1010-6 22

= 22500 n/C

+10 C

+10 C -10 C

A

B C

|EB| = kqr2

=

9109·1010-6 ( 22+22 )2

= 11250 n/C

|EC| = kqr2

=

9109·1010-6 22

= 22500 n/C

Now sketch in the field vectors:

EA

EB EA

EC

Now sum up the field vectors:

E = + + EA

EB

EC = 22500i + 11250 cos 45i + 11250 sin 45j - 22500j

= 30455i - 14545j (n/C)

Topic 6.2 ExtendedB – Electric field

What is the electric field along the bisector of a distant dipole having a charge separation d?

A dipole is an equal positive and negative charge in close proximity to one another:

+q

-qx

d2d2

r

r

Both field vectors have the same length: |E| = kqr2

Note that the horizontal components cancel, and the vertical components add: |ETOTAL| = 2|E| sin .But sin = (d/2)/r = d/2r so that

|ETOTAL| = 2|E| sin =kqdr3

But r = (d/2)2 + x2 . Since x >> d, r x and we have

EDIPOLE =kqdx3

Electric field far from a dipole

FYI: The dipole electric field drops off by 1/x3 rather than inverse square like a "monopole."