GAUSS'S LAW FOR SYMMETRIC CHARGE DISTRIBUTIONS ...

sentencecopyElectronics - Devices

Oct 13, 2013 (4 years and 26 days ago)

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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



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

=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
EXAMPLE:
What is the magnitude of the electric field near an infinitely large, uniformly
charged plane with charge per unit area of
σ

=
=
=