27: The Electric Field and Surface Charge at a Conductor

INTRO:
Learning Goal:
To understand the behavior of the electric field at the surface of a conductor, and its relationship to surface charge on the conductor.
A conductor is placed in an external electrostatic field. The external field is uniform before the conductor is placed within it. The conductor is completely isolated from any source of current or charge.

-----------------------------------------------------------------------------------------------------

PART A:
Which of the following describes the electric field inside this conductor?

• It is in the same direction as the original external field.
• It is in the opposite direction from that of the original external field.
• It has a direction determined entirely by the charge on its surface.
• It is always zero.

SOLUTION:
<< explanation to be added >>
It is always zero

NOTE:
The net electric field inside a conductor is always zero. If the net electric field were not zero, a current would flow inside the conductor. This would build up charge on the exterior of the conductor. This charge would oppose the field, ultimately (in a few nanoseconds for a metal) canceling the field to zero.
-----------------------------------------------------------------------------------------------------
PART B:
The charge density inside the conductor is:

• 0
• non-zero; but uniform
• non-zero; non-uniform
• infinite

SOLUTION:
<< explanation to be added >>
0

NOTE:
You already know that there is a zero net electric field inside a conductor; therefore, if you surround any internal point with a Gaussian surface, there will be no flux at any point on this surface, and hence the surface will enclose zero net charge. This surface can be imagined around any point inside the conductor with the same result, so the charge density must be zero everywhere inside the conductor. This argument breaks down at the surface of the conductor, because in that case, part of the Gaussian surface must lie outside the conducting object, where there is an electric field.
-----------------------------------------------------------------------------------------------------

PART C:

Assume that at some point just outside the surface of the conductor, the electric field has magnitude E and is directed toward the surface of the conductor. What is the charge density η on the surface of the conductor at that point?
Express your answer in terms of E and ϵ₀.

SOLUTION:
<< explanation to be added >>
η=-E⋅ε₀

HINTS:

1) How to approach the problem.​

Use Gauss's law with a short and flat cylindrical surface (picture a coin) with one end just below (inside) and the other just above (outside) the surface of the conductor.

2) Calculate the flux through the top of the cylinder​

Using a flat cylinder with large top and bottom each of area A just above and just below the surface of the conductor, find the flux Φ_top generated through the top surface of the cylinder by the electric field of magnitude E that points into the surface.
Express your answer in terms of A, E, and any needed constants.
→ φ_top = -EA

3) Calculate the flux through the bottom of the box​

Using a flat cylinder with large top and bottom of area A just above and just below the surface of the conductor, find the flux Φ_bot generated through the bottom surface of the cylinder by the electric field inside the conductor (keep in mind that positive flux is outward through the cylinder's surface, which is downward into the conductor).
Answer in terms of A, E, and any needed constants.
→ Φ_bot = 0

4) What is the charge inside the Gaussian surface?​

Find the net charge q_in inside this Gaussian surface.
Express your answer in terms of the charge density η and other given quantities.
→ q_in = ηA

5) Apply Gauss's law​

Now apply Gauss's law, neglecting any contribution to the flux due to the very short sides of the cylinder. Gauss's law states that ϵ₀⋅Φ_E=q_in. The area A should cancel out of your result.
→ η = -E⋅ε₀