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(III) Repeat Problem 19 assuming the charge density ρE increases as the square of the distance from the center of the sphere, and ρE = 0 at the center.
We assume the total charge is still Q, and let ρE = kr2. We evaluate the constant k by calculating the total charge, in the manner of Example 22-5.
(a) The electric field outside a charged, spherically symmetric volume is the same as that for a point charge of the same magnitude of charge. Integrating the electric field from infinity to the radius of interest gives the potential at that radius.
(b) Inside the sphere the electric field is obtained from Gauss’s Law using the charge enclosed by a sphere of radius r.
Integrating the electric field from the surface to r < r0 gives the electric potential inside the sphere.
(c) To plot, we first calculate
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