1. Can functions V1(R,θ) = C1Rcosθ and V2 =(R,θ) = C1R2 cosθ, where C1 and C2 are arbitrary constants, be solutions of Laplace's equation in spherical coordinates? Explain.
2. A point charge Q exists at a distance d above a large grounded conducting planes.
Determine
-the surface charge density ρs.
- the total charge induced on the conducting plane.
3. Two dielectric media with dielectric constants ε1 and ε2 are separated by a plane boundary at x = 0, A point charge Q exists in medium 1 at distance d from the boundary
a - Verify that the field in medium 1 can be obtained from Q and an image charge -Q 1, both acting in medium1.
b- Verify that the field in medium 2 can be obtained from Q and an image charge + Q2 coinciding with Q, both acting in medium 2.
c- Determine Q1 and Q2. (Hint : Consider neighboring points P1 and P2 in media 1 and 2, respectively, and require the continuity of the tangential component of the E-field and of the normal component of the D field)