Rutgers Physics 385 Electromagnetism I (Fall15/Gershtein)
Homework 7 - Due October 29, 2015 in class
1) The surface charge density on a sphere of radius R is given by
σ = k (cos(2θ) + 1/3) , where k is a constant.
Find the potential inside and outside of the sphere. What is the electric field at the center
of the sphere? Could you have guessed its value?
2) Solve Laplace's equation by separation of variables in cylindrical coordinates, assuming there
is no dependence on z, i.e. V(ρ,φ,z) = V(ρ,φ) = R(ρ)Φ(φ).
Hints:
for the R(ρ) solution (i.e. radial), try the power law
make sure that you find all solutions for the radial equation (you should have two independent solutions)
recall that the potential of a line charge is -λ ln(r/b) / (2πε0), where r is the
distance to the line charge and b is the distance from the line to a point where potential is taken to be zero.
The line charge problem has cylindrical symmetry and therefore should be described by your solution.
Basically, make sure you have a logarithmic term :)
3) Find the potential outside an infinitely long metal pipe, of radius R, placed at right angles to an otherwise uniform electric field
E. Find the surface charge density distribution on the pipe.
4) A rectangular pipe, running parallel to the z-axis (from minus infinity to plus infinity),
has three grounded metal sides, at y=0, y=a and x=0. The fourth side, at x=b, is maintained
at a specified potential Vo(y).
a) develop a general formula for the potential within the pipe
b) Find the potential explicitely for the case Vo(y)=Vo (i.e. constant).