I have four questions based on this principal that I'm struggling with.
1. The Potential at the surface of a sphere of radius R is given by V = kcos(3θ). K is a constant. (Assume no charge inside or outside the sphere).
a) Find the potential inside and outside the sphere.
b) Determine the surface charge density σ(θ) on the sphere.
2. The potential at the surface of a sphere V(t) is specified and there is no charge inside or outside the sphere. Show that the charge density on the sphere is given by.
σ(θ) = ε/2R Ʃ(2l+1)^2(Cl)(Pl)(cosθ)
The sum being from l=0 to l=∞
Cl = ∫V(θ)(Pl)(cosθ)(sinθ)dθ
The integral being from 0 to π
3. Rectangular pipe (running parallel to the z-axis (from -∞ to +∞) has three grounded metal sides, at y=0 and x=0. The fourth side at x=b, is maintained at a specified potential V(y).
a) Determine a general formula for the potential in the pipe.
b) Determine the potential in the pipre for the case V(y)=V=Constant
4. A box in the shape of a cube has 6 conducting sides of length a. Five of its sides are grounded. Assume the cube to have a corner at the origin and faces at x=0 x=a y=0 y=a z=0 and z=a. Only the top size z=a is insulated from the others and held at a constant potential V. Determine the potential inside the box.