Fall 2017 Homework Assignments
A dipole p = j p is placed at the origin. (j is the unit vector in the y-direction.) A point charge Q is placed a distance R away from the origin, on the x-axis. Find the forces and the torques on the system and verify that the total force and torque on the system is zero. (Hint: To find the force and the torque on the dipole, model it as two point charges a distance d apart, then evaluate the proper limit as d approaches zero.)
We are interested in finding the potential in the subspace defined by 0 < x < ∞ , 0 < y < ∞ , -∞ < z < ∞ . The boundary at y = 0 is kept at zero potential. The boundary at x = 0 is kept at a constant potential Vo. Using the Green's Function methodology, find the integral which gives the potential at arbitrary points inside the subspace.
A conducting grounded spherical shell of radius R is centered on the origin. A uniformly charged thin ring with radius r and total charge Q is centered around the z-axis, outside the sphere, at a distance D from the origin. Find the image of the ring-charge inside the sphere, and use that to determine the electric potential at all points on the z-axis.
Find the potential inside the cylindrical region -H/2 < z < H/2 and R1 < ρ < R2 for the configuration in the figure. The surfaces at z=±H/2 and at ρ=R2 are at zero potential. The surface at ρ=R1 has potential Vo for |z|<h/2 and zero for h/2<|z|<H/2 . Extra credit: Obtain a computer plot of this potential as a function of ρ and z. Assign proper numerical values to the parameters so that you obtain a pretty picture. (We may hold a beauty contest among participants.)
A dipole of magnitude p and pointing in the z-direction is placed on the z-axis, at z=a. The spherical region with radius R, centered at the origin, has permittivity ε. Rest of the space has permittivity εo. Find the expansion coefficients for the potential for r<R and r>R. Comment on the terms associated with l=0 and l=1. Extra credit: Repeat the above problem when the dipole is pointing in the x-direction (but still on the z-axis). Note that this is now a much more difficult problem. (The geometry no longer has azimuthal symmetry.) Therefore, do not start this part until after the exam.
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.