HW 2 Phys5405 f05 due Sept. 8
1) Charge is arranged on the surface of a sphere of radius R centered at the origin as follows:
σ
=
σ
0
, if
θ > α
and
σ
= 0, otherwise. Here
θ
is the usual polar angle, 0
≤
θ
≤
π
.
a)Find
E
at any point on the z axis. Hintbe careful about positive and negative z and

z

greater or less than R.
b)In particular obtain
E
at z=R, z=0, and z=

R
c)Take R=1 m and plot E(
z
)
/
(
σR
2
/
2
0
) as z varies from z=

2 m to z= 2 m for cos
α
=
0.999, 0., 0.999.
Use of a package like Mathematica makes this easy. Otherwise you can write a fortran or
C++ program and make the plots by hand. Be careful about the point z=

R. Examine
the plots and see if they make sense. They provide more information than just a formula.
d)For
E
at z=R and z=

R, calculate the limits as
α
→
0.
e)When
α
= 0 you can use Gauss’s law to calculate
E
anywhere inside or outside the
spherical surface. However, if you used Gauss’s law to calculate
E
at R, you would need to
use a Gauss surface that coincided with the charges. If you did so and included the charges
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 Spring '10
 Blecher
 Charge, Magnetism, Gauss’s Law, Gauss surface

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