# Conductors (U. of Washington)

1. We can use the fact that the net electric field is zero inside a conductor to conclude several things about the field at the surface of conductors.  These are called boundary conditions on conductors.

a. If a system involving conductors is in equilibrium, are any of the charges inside or on the surface of the conductor moving? Explain.

b. Point A is at the surface of a conductor (not inside the conductor). Given your answer above, which (if any) of the vectors shown could represent the net electric field at point A? Explain.

c. What does your answer suggest about the component of the net electric field that 13 parallel to the surface of the conductor?

d. What does your answer suggest about the component of the net electric field that is perpendicular to the surface of the conductor?

e. Suppose that point A was on a flat surface of a conductor with surface charge density $+sigma$. A small cubical Gaussian surface with side length $l$

surrounds point A.

i. If $l$ is infinitesimally small, we are measuring the electric field extremely close to the surface of the conductor. In which direction is the electric field at point A?

ii. Using Gauss’s law and your answer above, find the parallel and perpendicular components of the net electric field at point A.

a) The source of electric fields are electric charges. An equilibrium in an electric system means constant electric fields over time. Since moving charges would imply changing electric fields over time, this means that in a system at equilibrium existent charges are not moving.

b) If the system is in equilibrium them this should mean charge at A (if it exists) is not moving on the surface of the conductor. All electric fields that have a component parallel to the surface of the conductor would move the charge on the conductor surface. (For charge A to move, one needs a force, and the origin of force is an electric field). Therefore just fields perpendicular to the surface of the conductor can exist. There are possible just fields A and E.

c,d) The parallel to the surface component of the electric field (for a conductor in equilibrium) is always zero. If there are electric charges in the system then there exists a normal (perpendicular) nonzero electric field.

e)  Electric fields are out of the positive charges and into negative charges. Thus for a positively charged conductor sheet the electric field exactly at point A is towards exterior of conductor perpendicular to the conductor. Inside the conductor the field is zero in all directions (equilibrium so that existent charges on the surface do not move into volume). Outside the conductor the flux of the electric field on the lateral sides 1 and 2 of the cube cancels each other (opposite electric fields on these surfaces). It remains just the flux on the surface 3. Gauss law is

$E(l/2)*S3 =Q_{inside}/epsilon$   with $Q_inside = sigma*l^2$ and $S3 =l^2$
Therefore

$E(l/2) = sigma/epsilon$ or said differently $E(l) = (sigma)/(2epsilon)$

When length tends to zero we have the field exactly at point A.

Therefore at point A the parallel to the surface component of the field is zero and the perpendicular component is $E =frac{sigma}{2epsilon}$