Please explain this problem to me thoroughly. A solid insulating sphere of radiu
ID: 1494782 • Letter: P
Question
Please explain this problem to me thoroughly.
A solid insulating sphere of radius R has a non-uniform charge density that varies with the distance r from the center according to the formula p(r) = p0(exp)(-r/b)/r^2 where b and p0 are constants.
A)What type of Gaussian surface would you use in this problem?
B) Derive an expression for the electric flux trough the Gausssian surface in terms of the electric field strength E(r).
C) Derive an expression for E(r) a distance r < R from the sphere’s center in terms of the constants p0 and b. Note: The volume element for a sphere shell of radius r and thickness dr is dV = 4(pi)(r^2)dr.
Explanation / Answer
1. A spherical gaussian surface would be used in this problem
2. For a gaussian surface of radius r
E(r).A = q(in)/epsilon
where A is the area of the gaussean surface
epsilon is the permittivity of free space
q(in) is the charge enclosed by the gaussian surface
3. For r < R
Charge enclosed, Qin = [integral][p0[(exp)(-r/b)]dV/r^2] = [integral][p0[(exp)(-r/b)]4pi*dr]
Qin = -[4pi*p0*b(exp)(-r/b)] from r = 0 to r = r
Applying limits
Qin = 4pi*p0*b*[1 - exp(-r/b)]
so, E(r)*4*pi*r^2 = 4pi*p0*b*[1 - exp(-r/b)]/epsilon
E(r) = p0*b*[1 - exp(-r/b)]/epsilon*r^2