PHYS 303 Final, Fall 1996

LIBRARY fee Physics 303 January 29, 1997 Physics Department Classical Electrodynamics Time:_3_hours Final Exam Choose two out of the three problems 1,2 and 3 and solve 4. 30 marks 1) Consider a conducting sphere of radius Ry, on which is placed a charge Qs In contact and concentric with this sphere is dielectric material having a dielectric constant € = 1+ 4% , that extends out to a radius R, . We wish to find the fields and charge densities generated everywhere. 1) Does # depend on 0 and 9? 2) Apply Gauss’ Law to find D, and #, for R; sr< R.. 3) Find D, and #, wherer>R,. 4) What are the fields D and £ atr< R,. 5) What is the polarization P atr=R,. 6) Deduce the charge density os at r = Ry. 7) Findo, atr= R». 8) Determine the potential of the conductor 6. 30 marks 2) | We shall examine the potential function , associated with a charge distribution shown in the figure, and given by : : L L P = Po for - SBStT- p= 0 for |z| > 5. Use symmetry arguments, and » =0 at z = 0 ( Indicate why !) plus other boundary conditions to solve Poisson’s (or Laplace’s ) equation in the three regions 1, 2, and 3. (Assume that there is no surface charge density in this problem ) 30 marks 3) Find the electrostatic energy of two identical small, charged conducting spheres at a large distance from each other. Then calculate the change in this electrostatic energy when a thin wire is used to connect the spheres electrically. 40 marks \ 4) Consider a sphere of radius R that has a charge uniformly distributed over its surface. Let o be the surface charge density. The sphere is spinning with a constant angular velocity about an axis Zi going through its center. The density of the surface Co8 current produced as a result of spinning is K = o ¥ where ¥ = woRsinO.9 is the velocity of the charge element at angle 6. The vector potential inside and outside the sphere satisfy Laplace’s equation. 4 is taken in the @ direction and independent of p: 4 =Ag(1,0) @. Determine expressions for 4 whenr > Rorr<R. The andetenaned constants in 4 may be solved with the help of 2 conditions: the continuity of 4 and the condition relating the tangential components of H at r=R. Compare your result with the well known problem of the potential produced by a uniformly magnetized sphere.