Applications of Gauss Law

Example of Gauss’ Law

• Consider a dipole with equal positive and negative charges.
• Imagine four surfaces S₁, S₂, S₃, S₄, as shown.
• S₁ encloses the positive charge. Note that the field is everywhere outward, so the flux is positive.
• S₂ encloses the negative charge. Note that the field is everywhere inward, so the flux through the surface is negative.
• S₃ encloses no charge. The flux through the surface is negative at the upper part, and positive at the lower part, but these cancel, and there is no net flux through the surface.
• S₄ encloses both charges. Again there is no net charge enclosed, so there is equal flux going out and coming in—no net flux through the surface.

Field At the Surface of a Conductor

• Imagine an electric field at some arbitrary angle at the surface of a conductor.
• There is a component perpendicular to the surface, so charges will move in this direction until they reach the surface, and then, since they cannot leave the surface, they stop.
• There is also a component parallel to the surface, so there will be forces on charges in this direction.
• Since they are free to move, they will move to nullify any parallel component of E.
• In a very short time, only the perpendicular component is left.

Field Inside a Conductor

• We can use Gauss’ Law to show that the inside of a conductor must have no net charge.
• Take an arbitrarily shaped conductor, and draw a gaussian surface just inside.
• Physically, we expect that there is no electric field inside, since otherwise the charges would move to nullify it.
• Since E = 0 everywhere inside, E must be zero also on the gaussian surface, hence there can be no net charge inside.
• Hence, all of the charge must be on the surface (as discussed in the previous slide).
• If we make a hole in the conductor, and surround the hole with a gaussian surface, by the same argument there is no E field through this new surface, hence there is no net charge in the hole.

Field Inside a Conductor

• We have the remarkable fact that if you try to deposit charge on the inside of the conductor...
• The charges all move to the outside and distribute themselves so that the electric field is everywhere normal to the surface.
• This is NOT obvious, but Gauss’ Law allows us to show this!

There are two ideas here

• Electric field is zero inside conductors
• Because that is true, from Gauss’

Law, cavities in conductors have E = 0

Charge Distribution on Conductors

• For a conducting sphere, the charges spread themselves evenly around the surface.
• For other shapes, however, the charges tend to collect near sharp curvature.
• To see why, consider a line of charge.

A Charge Inside a Conductor

2. What will happen when we add a charge inside a conductor?
1. E field is still zero in the cavity.
2. E field is not zero in the cavity, but it is zero in the conductor.
3. E field is zero outside the conducting sphere.
4. E field is the same as if the conductor were not there (i.e. radial outward everywhere).
5. E field is zero in the conductor, and negative (radially inward) outside the conducting sphere.

Conducting

sphere

Spherical

cavity

Positive point

charge

Use Gauss’ Law to Find Out

Is E = 0 in the cavity?

Gaussian Surface

No, because there is charge enclosed (Gauss’ Law).

Is E = 0 in the conductor?

Yes, because as before, if there were an electric field in the conductor, the charges would move in response (NOT Gauss’ Law).

If we enlarge the gaussian surface so that it is inside the conductor, is there any net charge enclosed?

It looks like there is, but there cannot be, because Gauss’ Law says E = 0 implies q_enc = 0!

How do we explain this?

There must be an equal and opposite charge induced on the inner surface.

E Field of Charge In Conductor

This negative charge cannot appear out of nowhere.

Where does it come from?

It comes from the outer surface (electrons drawn inward, attracted to the positive charge in the center). Therefore, it leaves positive charge behind.

The net positive charge that appears conductor is exactly the same as the original charge in the center, so what do the field lines look like?

By spherical symmetry, the positive shell of charge acts like a point charge in the center, so field is the same as the field of the original point charge.

This negative charge acts with the inner charge to

make the field radial inside the cavity.

E Field of Charge In Conductor

This negative charge cannot appear out of nowhere.

Where does it come from?

It comes from the outer surface (electrons drawn inward, attracted to the positive charge in the center). Therefore, it leaves positive charge behind.

The net positive charge that appears conductor is exactly the same as the original charge in the center, so what do the field lines look like?

By spherical symmetry, the positive shell of charge acts like a point charge in the center, so field is the same as the field of the original point charge.

This negative charge acts with the inner charge to

make the field radial inside the cavity.

Back to the Previous Question

3. What will happen when we add a charge inside a conductor?
1. E field is still zero in the cavity.
2. E field is not zero in the cavity, but it is zero in the conductor.
3. E field is zero outside the conducting sphere.
4. E field is the same as if the conductor were not there (i.e. radial outward everywhere).
5. E field is zero in the conductor, and negative (radially inward) outside the conducting sphere.

Conducting

sphere

Spherical

cavity

Positive point

charge

E Field of Charge In Conductor

It induces an off-center charge distribution on the

Inner wall.

Note that the field lines distorted, so they remain perpendicular to the inner wall. What happens to the outer positive charge distribution?

Draw a gaussian surface inside the conductor to find out.

The net charge enclosed is zero, so E = 0, which we already knew because it is inside the conductor. The inner charge is shielded by the induced charge distribution, so the outer charges will be evenly distributed.

What happens when we move the inner charge

off-center?

Field Lines and Conductors

4. Why cannot the drawing of the conductor and its field lines be correct?
1. The field lines are in both directions, coming to and leaving from the conductor.
2. One of the field lines loops around, with both end points on the conductor.
3. The field lines are not all perpendicular to the surface of the conductor.
4. All of the above.
5. B and C only.

Other Geometries

• Always use the symmetry of the problem to determine what shape to make your gaussian surface.
• Here is a plate (plane) geometry, where the charges are evenly distributed on a flat surface. If the total charge on the plate is Q, and the plate has a total area A_tot, then the surface charge density is

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• The E field is everywhere perpendicular to the plate (again, if not, the charges will move until the part parallel to the surface is nullified). What is ?
• Use a gaussian surface that is parallel to on the sides (so no flux through side surfaces), and closes inside the conductor (no flux through that end).
• On the remaining side, the area vector is parallel to the E field, so or

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Conducting Surface

Line of Charge

• In the previous chapter, we calculated the E field on the axis of a line of charge, but with Gauss’ Law we can now handle finding E off the line axis.
• Here is a line geometry, where the charges are evenly distributed on a long line. If the total charge on the line is Q, and the line has a total length L_tot, then the linear charge density is

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• The E field is everywhere perpendicular to the line (again, if not, the charges will move until the part parallel to the line is nullified).
• Use a cylindrical gaussian surface that is parallel to on the top and bottom (so no flux through those surfaces), and is perpendicular to elsewhere.
• The area vector is parallel to , and the total area is 2πrh so

or

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Line of Charge

Symmetry

5. A charged sphere has an electric field just like that of a point charge of the same total charge, located at its center. What is the electric field of a long conducting cylinder like?
1. Also like that of a point charge at its center.
2. Like a circular ring of charge at its center.
3. Like a line charge along the cylinder axis.
4. Cannot tell from the information given.

Uniform Sphere of Charge

• Here is a spherical geometry, where the charges are evenly distributed throughout the volume. If the total charge in the sphere is Q, and the sphere has a radius R, then the volume charge density is

​

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• By symmetry, the E field is everywhere radial from the center of the sphere.
• Use a spherical gaussian surface, which is perpendicular to everywhere.
• The area vector is parallel to , and the total area is 4πr² so when the gaussian surface radius is r < R, then

or

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• When r>R, then the charge enclosed is just Q, so

or

Coulomb’s Law again

Nonconducting Sheet

• A nonconducting sheet with a uniform surface charge density has the same geometry as for the conducting plate, so use the same gaussian surface.
• The only difference is that now one end cannot close in a conductor, so there is electric flux out both ends.
• As you may expect, the resulting electric field is half of what we got before.

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Sheet of Charge

Two Parallel Conducting Plates

• When we have the situation shown in the left two panels (a positively charged plate and another negatively charged plate with the same magnitude of charge), both in isolation, they each have equal amounts of charge (surface charge density σ) on both faces.
• But when we bring them close together, the charges on the far sides move to the near sides, so on that inner surface the charge density is now 2σ.
• A gaussian surface shows that the net charge is zero (no flux through sides — dA perpendicular to E, or ends — E = 0). E = 0 outside, too, due to shielding, in just the same way we saw for the sphere.

Two Parallel Nonconducting Sheets

• The situation is different if you bring two nonconducting sheets of charge close to each other.
• In this case, the charges cannot move, so there is no shielding, but now we can use the principle of superposition.
• In this case, the electric field on the left due to the positively charged sheet is canceled by the electric field on the left of the negatively charged sheet, so the field there is zero.
• Likewise, the electric field on the right due to the negatively charged sheet is canceled by the electric field on the right of the positively charged sheet.

• The result is much the same as before, with the electric field in between being twice what it was previously.

Spherical Symmetry

Spherical shell

We earlier said that a shell of uniform charge attracts or repels a charged particle that is outside the shell as if the shell’s charge were concentrated at the center of the shell. We can now prove this using Gauss’ Law.

We also said that a shell of uniform charge exerts no electrostatic force on a charged particle that is located inside the shell. Again, Gauss’ Law can be used to prove this.

Summary

• Electric flux is the amount of electric field passing through a closed surface.
• Flux is positive when electric field is outward, and negative when electric field is inward through the closed surface.
• Gauss’ Law states that the electric flux is proportional to the net charge enclosed by the surface, and the constant of proportionality is ε₀. In symbols, it is

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• There are three geometries we typically deal with:

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Thanks