A nonuniform, but spherically symmetric, insulating sphere of charge has a charge density ρ(r)ρ(r) given as follows: ρ(r)=ρ0(1−4r/3R) for r≤R ρ(r)=0 for r≥R where ρ0 is a positive constant. This means that the amount of charge per unit volume varies with distance from the center of the sphere. Hint: The charge enclosed in a given spherical shell of inner radius R1 and outer radius R2 is Qenc=∫R2R1 ρ(r)dV where dV=4πr^2dr (in
A nonuniform, but spherically symmetric, insulating sphere of charge has a charge density ρ(r)ρ(r) given as follows:
| ρ(r)=ρ0(1−4r/3R) | for r≤R |
| ρ(r)=0 | for r≥R |
where ρ0 is a positive constant. This means that the amount of charge per unit volume varies with distance from the center of the sphere.
Hint: The charge enclosed in a given spherical shell of inner radius R1 and outer radius R2 is Qenc=∫R2R1 ρ(r)dV where dV=4πr^2dr (in other words, you have to integrate over each spherical shell of radius rr from the inner radius to the outer radius). Notice that if the charge is uniform, so that ρρ is a constant, and you have a solid sphere with inner radius zero and outer radius R (i.e. a normal sphere) you can pull ρρ out of the integral and it reduces to the simple expression Qenc=ρ4/3πR^3. For a non-uniformly charged sphere, however, as in this problem, you have to use the full integral to find the charge enclosed in a given region.
A) Obtain an expression for the electric field in the region r≤R.
B) Find the value of rr at which the electric field is maximum.
C) Find the value of that maximum field.
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
