GAUSS’S LAW FOR SYMMETRIC CHARGE DISTRIBUTIONS (19.10)

Recall Gauss’s Law:

0

inside

surface

closed

over

e

ε

q

AdE=⋅=Φ

∫

r

r

Ad

r

is a vector normal to surface with magnitude equal to the area of the surface

element

How do we use Gauss’s Law? Two Ways:

•

0

inside

e

ε

q

=Φ

o relates flux to charge inside for surface of ANY shape

•

0

inside

surface

closed

over

ε

q

AdE=⋅

∫

r

r

o gives a way to calculate

E

r

潲 SPECIAL cases

mostly useful if we normal component of

E

r

猠捯湳瑡湴癥爠灡牴映

瑨攠獵牦慣攠慮搠捡渠扥慣瑯牥搠潵琠潦⁴桥湴敧牡氠

Using Gauss’s law to calculate Electric Field

E

r

• Must know direction of electric field from symmetry of problem

o radial (spherical symmetry) for point charge

o radial (cylindrical symmetry) for a long line of

charge

o uniform for a large flat sheet of charge

• Must choose Gaussian surface that allows us to calculate

∫

⋅=Φ

surface

closed

over

e

AdE

r

r

o Must be able to factor

E

r

畴映flux integral in region of space where we

want to find electric field

Two cases for which we can evaluate

∫

⋅

surface

AdE

r

r

for all or part of surface

•

E

r

uniform and perpendicular to part or all of Gaussian surface

o then flux is

∫

=⋅AEAdE

n

r

r

for that part of the surface

•

E

r

parallel (tangent) to part of the Gaussian surface

o then

0=⋅AdE

r

r

for that part of surface → no contribution to total flux

EXAMPLE:

(a) Find the electric field INSIDE and OUTSIDE of a uniformly charged

insulating sphere with radius a and total charge Q.

(b) Plot the magnitude of

E

r

慳畮捴楯渠潦†摩獴慮捥ar from sphere centre.

(c) Calculate the electric potential

(

)

rV

inside and outside the sphere.

1st: Look at

ar<

(inside of sphere)

• Draw Gaussian surface, radius r, same centre

• By symmetry,

E

r

is

⊥

⁴漠䝡畳獩慮畲晡捥癥特睨敲攠

• Normal component

n

E

is uniform over the surface

2nd: Look at

ar>

(outside of sphere)

• Draw Gaussian surface, radius r, same centre

• By symmetry,

E

r

is

⊥

⁴漠䝡畳獩慮畲晡捥癥特睨敲=

• Normal component

n

E

is uniform over the surface

EXAMPLE:

What is the magnitude of the electric field at a perpendicular distance r from

an infinitely long, uniformly charged rod with charge per unit length of

λ

㼠

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

EXAMPLE:

What is the magnitude of the electric field near an infinitely large, uniformly

charged plane with charge per unit area of

σ

㼠

=

=

=

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