Unit 1 Day 11: Applications of Gauss’s Law

• Spherical Conducting Shell

• A Long Uniform Line of Charge

• An Infinitely Large, Thin Plane of Charge

• Experimental Basis of Gauss’s Law

A Thin Spherical Conducting Shell of Radius r0

• Charge is uniformly distributed

• E-Field is symmetric at all points

• Outside the shell the E-Field penetrates surface A1 perpendicular to the surface

• Inside the shell (r < r0), the E-Field does not penetrate A2, and E=0, because Q=0

200

2

414rQEorQrEdAE

Solid Sphere of Charge• Outside the Sphere (r > r0)

• Inside the Sphere (r < r0 )

but Qencl is not the total charge Q but a percentage of it, by volume:

200

2

414

rQEorQrEAdE enclencl

QrrQencl 30

3

orQrEAdE 0

24

204

1rQE

rrQE 3004

1

Solid Sphere of Charge

rrQE 3004

1

204

1rQE

rrQE 3004

1

A Uniform Line of Charge

• A very long straight wire, possessing a uniform charge density:

• The Gaussian Surface is a cylinder where the E-Field is perpendicular at all points

dldQ

00

2

lQRlEdAE encl

RE

021

An Infinitely Large, Thin, Plane of Charge

• Assume a uniform charge density

• Choose a Gaussian Surface as a small closed cylinder whose axis is perpendicular to the plane

• The E-Field is directed perpendicular to the plane on both sides & uniform over the end cap of the cylinder of area A

dAdQ

00

2

AQEAdAE encl

02

E

Experimental Basis of Gauss’s & Coulomb’s Laws

• Gauss’s Law requires that any net charge must reside on the surface of a conductor

• The charge will flow to the outside

• Test the outside of the can for charge

• Test the ball after touching inside of can & withdrawn: charge will be zero