1. If is a solution of Laplace’s equation, show that one or more derivatives of with respect to rectangular coordinates also satisfy Laplace’s equation.

2. Using the property of solutions to Laplace’s equation show that there cannot be an electrostatic field inside a hollow conductor unless there is a charge in that region.

The inner surface of the conductor which encloses the hollow is an equipotential. Let the potential be .If there are no charges in the hollow, the potential there must satisfy Laplace’s equation.  We know that solutions to Laplace’s equations have no local minima or maxima. Thus the potential inside the hollow must be the same as that on its boundary, i.e., the potential has a constant value . Thus the electric field inside the hollow is zero.

3. A conductor has a hollow scooped out inside. The conductor carries a charge and the potential of the conductor is with respect to its zero at infinity. Show that the potential inside the hollow is also

Conductor is an equipotential. Thus the inner surface of the hollow is also at a potential . The potential inside the hollow must have the same value of the potential because (i) being a constant, it satisfies Laplace’s equation and (ii) it satisfies Dirichlet boundary condition on the surface. By uniqueness theorem, this is the only possible solution.

4.Consider two point conductors separated by a distance of 10m. One of the conductors is maintained at a potential of 100 V while the other maintained at 200 V.  Determine the potential function and the corresponding electric field .

Let the first conductor be located at the origin and the second at x=10.   On the entire x axis other than at these two points, Laplace’s equation is applicable, which has the solution . Inserting boundary conditions we have, so that the potential is Volts. The electric field is V/m.