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Chapter 22

Gausss Law 22.1, 22.2 The Electric Flux

!

"E The electric flux

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"E through a surface is defined as

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"E =r E ext #

r A = Eext A cos $( )

where Ar

= area vector of the surface, perpendicular to the surface.

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r E ext = electric field at the surface. = angle between

!

r E ext and A

r.

more generally,

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"E #r E ext $d

r A %

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22.3 Gausss Law Gausss law is used to calculate the electric field generated at point P by a charge distribution that has enough symmetry in it. Here you imagine a closed 3-dimensional surface (i.e., sphere, cylinder,) and consider the following integral:

o

encQAdE!

="#rr

Comments:

1. the surface you imagine is not there and it is called a Gaussian surface.

2. the circle in the integral sign means we consider a closed surface (Gaussian surface).

3. Er

is the electric field at the Gaussian surface. 4. Ad

ris an area vector with a magnitude equal to the

area of the Gaussian surface and a direction perpendicular to the Gaussian surface (outward).

5. encQ is the total charge inside or enclosed by the Gaussian surface.

6. Gausss law is useful and practical only when there is enough symmetry in the charge distribution. Use the following table as a guide:

7. The left-hand side of Gausss law is called the electric flux through the Gaussian surface.

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Given geometry Gaussian surface used

Point charge sphere

sphere sphere

! long wire cylinder

! long cylinder cylinder

! large sheet Cylinder

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22.4 Applications of Gausss Law to Various Charge Distributions The best way to illustrate Gausss law is for me to solve on the board the following examples: 1. A solid insulating sphere of radius a has a non-uniform charge density that varies with r according to = Ar2, where A is a constant and r < a is measured from the center of the sphere. What is the total charge Q of the sphere? 2. Use Gausss law to find the electric field everywhere in space due to

(a) a point charge q (b) an insulating sphere of radius a with total charge Q (charge density ). (c) an infinitely long wire with charge per unit length , (d) an infinitely long conducting cylinder of radius a and charge per unit

length , (e) an infinitely long insulating cylinder of radius a and charge per unit

volume , (f) an infinitely long sheet with charge per unit area , (g) paralle plates with charge per unit area on one plate and and charge per

unit area - on the other plate, (h) a conducting sphere of radius a with total electric charge Q ,

3. A point charge +q is located at the center of a hollow conducting sphere of inner radius a and outer radius b. The net charge on the sphere is Q. (a) Use Gausss law to find the electric field magnitude everywhere in space. (b) What is the charge per unit area on the inner surface of the sphere with radius a? (c) What is the charge per unit area on the outer surface of the sphere with radius b?

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22.5 Charges on Conductors in Electrostatic Equilibrium The charges are not in motion in a conductor that is in electrostatic equilibrium.

0=Er

every where inside the body of the conductor. Any net charge resides on the surface of the

conductor

Just outside the conductor, the electric field is perpendicular to the surface and with magnitude equal to

oE

!"

=r

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A conducting box (a Faraday cage) immersed in a uniform electric field. The electric field of the induced charges on the box combines with the uniform external electric field to give zero total electric field inside the box. The person inside the Faraday cage is protected from the powerful electric discharge.