# Conductors in Electrostatic Equilibrium

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Conductors in Electrostatic Equilibrium
AP Physics C Mrs. Coyle

Gauss’s Law Permittivity of free space:
ε0 = x C2 / (N m2)

Insulators vs Conductors
In an insulator, excess charge is not free to move. In conductors the electrons are free to move.

Electrostatic Equilibrium of Conductors
Electrostatic Equilibrium for a conductor – no net motion of charge within a conductor. Most conductors, on their own, are in electrostatic equilibrium. Ex: in a piece of metal sitting by itself, there is no “current.”

Characteristics of Conductors in Equilibrium
The E-field is zero at all points inside a conductor (regardless if it is hollow or solid). If an isolated conductor carries excess charge, the excess charge resides on its surface. The E-field just outside a charged conductor is perpendicular to the surface and has magnitude σ/ε0, where σ is the surface charge density at that point. On an irregularly shaped conductors the surface charge density is biggest where the conductor is most sharp.

If the conductor is placed in an electric field at first there is a movement of electrons(current) but eventually the movement stops and their is equilibrium. If the E was not zero inside the conductor the movement would continue and there would not be equilibrium.

Note: Inside the cylinder there are no electric field lines.

Ex 1: Point charge Inside a Spherical Metal Shell
A -5.0μC charge is located as shown in Fig a). If the shell is electrically neutral, what are the induced charges on its inner and outer surfaces? Are those charges uniformly distributed? What is the E-field pattern?

Ex 1: Solution Strategy Since the shell is electrically neutral, E=0 inside the shell. Take a Gaussian surface inside the shell. This Gaussian surface must encompass an enclosed charge of zero because E=0 inside the conductor. The point charge is –5μC so since the net charge is zero: –5μC + x =0  x= 5μC . This x is the charge on the inside surface of the shell. Since the shell is neutral the outside surface of the shell must have a charge of –x=-5μC

Ex 1: Solution Strategy cont’d
Since the point charge is not in the center of the spherical shell but off-centered, there will be more positive charges closer to the point charge. The charge distribution in the inner wall of the shell will be more dense closer to the point charge. The field lines between the point charge and the shell will be closer together nearest to the point charge. However, in the outer surface of the shell the negative charges will be evenly distributed. This is the case no matter where inside the shell, the point charge is located. The field lines are shown in figure b) E-lines are always perpendicular to the conductor surface.

Week 04, Day 2 Hollow Conductors Charge placed INSIDE induces balancing charge ON INSIDE Class 09 11

Week 04, Day 2 Hollow Conductor A charge placed OUTSIDE induces charge separation ON OUTSIDE surface. Class 09 12

Ex 2: Sphere inside a Spherical Shell
A solid insulating sphere of radius a carries a uniformly distributed charge, Q. A conducting shell of inner radius b and outer radius c is concentric and carries a net charge of -2Q. Find the E-field in regions 1-4 using Gauss’s Law. Find the charge distribution on the shell when it is in electrostatic equilibrium.

Example #31 Consider a thin spherical shell of radius 14.0 cm with a total charge of 32.0 μC distributed uniformly on its surface. Find the electric field 10.0 cm and 20.0 cm from the center of the charge distribution.

Ex. #35 A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00 μC. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and (b) the total electric flux through the cylinder.

Ex. #43 A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find the charge density of each face of the plate and the total charge on each face.

Ex. #47 A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ. From this information, use Gauss’s law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance r from the axis.