(a) Using the law of cosines, show that Eq. 3.17 can be written as follows: whe...
(a) Using the law of cosines, show that Eq. 3.17 can be written as follows:
where r and θ are the usual spherical polar coordinates, with the z axis along the line through q. In this form, it is obvious that V = 0 on the sphere, r = R.
(b) Find the induced surface charge on the sphere, as a function of θ. Integrate this to get the total induced charge. (What should it be?)
Reference equation 3.17
Step-by-Step Solution
Surface charge density on a solid is the total amount of charge per unit area.
The expression for potential using law of cosines as follows:
Here, r and 
are the spherical polar coordinates, q is the charge, 
is the permittivity, and R is the radius of sphere.
(a)
From the fig,
We know that,
Substitute 
for
.
The potential of this configuration is,
Substitute 
for
, 
for r, and 
for
.
Substitute 
for b.
The potential is zero when r=R.
Therefore, the potential is zero when r=R.
(b)
Induced surface charge density is,
But 
Substitute
for V.
Induced surface charge is,
Substitute 
for
.
For a>R, 
Substitute 
for 
and 
for
.
(c)
The force of image charge
on q is,
for b and 
for
. 
The work done is,
Substitute 
for F.
Therefore, the energy of this configuration is
.
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