
Concept explainers
A radial electric field distribution in free space is given in spherical coordinates as:
where p0, a, and b are constants. (a) Determine the volume charge density in the entire region (0 < r < �) by the appropriate use of V, D = pv. (b) In terms of given parameters, find the total charge, Q, within a sphere of radius r where r>b.

(a)
The volume charge density in the entire region (
Answer to Problem 3.20P
The volume charge density is given by
:
Explanation of Solution
Given information:
The radial electric field distribution in free space is given as:
Concept used:
The volume charge density
We know that
So, for each radius values we can obtain the corresponding flux density values and hence charge density from Gauss's law.
Calculation:
We are given electric field distribution in free space as:
We have spherical coordinate coordinate system here. The divergence of flux density
Here only radial component,
By Gauss's law:
Hence,
Substituting
So, electric flux densities are given by:
Hence, we can summarise the charge density values as:

(b)
The total charge, Q, within a sphere of radius r where r >b.
Answer to Problem 3.20P
The total charge
Explanation of Solution
Given information:
The electric field distribution in free space is given in spherical coordinates as:
From the previous part a), we have volume charge densities as:
Concept used:
We have to integrate the volume charge density over the volume of sphere of radius b using the piece-wise distribution given.
Calculation:
We have the volume charge densities as:
So, the enclosed charge is given by:
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Chapter 3 Solutions
Engineering Electromagnetics
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