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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