Question

A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ(r) given as follows:

ρ(r)=ρ0(1−r/R)	for r≤R
ρ(r)=0	for r≥R

where ρ0=3Q/πR3 is a positive constant.

Part A

Find the total charge contained in the charge distribution.

Express your answer in terms of some or all of the variables r, R, Q, and appropriate constants.

Part B

Obtain an expression for the electric field in the region r≥R.

Express your answer in terms of some or all of the variables r, R, Q, and appropriate constants.

Part C

Obtain an expression for the electric field in the region r≤R.

Express your answer in terms of some or all of the variables r, R, Q, and appropriate constants.

Part D

Find the value of r at which the electric field is maximum.

Express your answer in terms of some or all of the variables r, R, Q, and appropriate constants.

Part E

Find the value of that maximum field.

Express your answer in terms of some or all of the variables r, R, Q, and appropriate constants.

Answer 1

A) We can consider an infinitely thin shell having thickness dr and surface area 4πr². The charge on that shell is

dQ = ρ(r)*A*dr = ρ0(1 - r/R)(4πr²)dr = 4π*ρ0(r² - r³/R)dr

which when integrated from 0 to r is

total charge = 4π*ρ0(r³/3 + r^4/(4R))

and when r = R our total charge is

total charge = 4π*ρ0(R³/3 + R³/4) = 4π*ρ0*R³/12 = π*ρ0*R³ / 3

and after substituting ρ0 = 3Q / πR³ we have

total charge = Q ◄

B) E = kQ/d²

since the distribution is symmetric spherically

C) dE = k*dq/r² = k*4π*ρ0(r² - r³/R)dr / r² = k*4π*ρ0(1 - r/R)dr

so

E(r) = k*4π*ρ0*(r - r²/(2R)) from zero to r is

and after substituting for ρ0 is

E(r) = k*4π*3Q(r - r²/(2R)) / πR³ = 12kQ(r/R³ - r²/(2R^4))

which could be expressed other ways.

D) dE/dr = 0 = 12kQ(1/R³ - r/R^4) means that

r = R for a min/max (and we know it's a max since r = 0 is a min).

E) E = 12kQ(R/R³ - R²/(2R^4)) = 12kQ / 2R² = 6kQ / R²