Consider a conducting (i.e.
metallic) region defined by two half-planes connected to the ground (i.e. at
zero potential Φ), as shown in the figure. These planes are perpendicular to
one another i.e. one is defined by x = 0 and y > 0, and the other by y = 0 and x
> 0. A charge q is located are position P = (a, a, 0) as shown. The region
where the charge resides is in the vacuum, and it is called here the first
quadrant.
Note that this is a 3D problem, not just 2D.
(a) Using the method of images find the scalar potential
Φ(r) in the first quadrant
at an arbitrary point r = (x, y, z).
(b) Verify that the potential you found in (a) cancels in the two half-planes.
(c) Find the force on the charge q induced by the planes.
A small conducting ring of radius a is located in the xy-plane centered at
the origin. A positive charge Q is place on the ring.
(a) Find the potential Φ on the z-axis at z > a and expand your
expression in powers of a/z.
(b) The potential Φ(r,θ) at a arbitrary points in space with r >
a can be
expanded in terms of Legendre polynomials, Φ(r,θ) = ∑n=0∞[Anrn
+ Bn/rn+1]Pn(cosθ).
Find the expansion coefficients.
For this problem, consider only the Coulomb interaction,
i.e. neglect gravity, magnetism, and radiation.
A point charge of mass m = 10-3 kg and charge q
= 10-6 C and an insulated, conducting spherical shell of mass M = 10-1
kg and radius R = 0.01 m orbit their common center of mass in circular orbits.
The distance between the center of the sphere and the point charge is d = 0.1
m. The spherical shell spins, so that the line from the center of the sphere to the
point charge always passes through the same point on the shell.
Find the total angular momentum of the system about the
center of mass.
(Give a numerical answer!)
Two conducting cones (0 < θ1
< θ2 < π/2) of infinite extend are separated by an
infinitesimal gap at r = 0.
Let V(θ1) = 0 and V(θ2) = V0. Find V
between the cones.
Laplacian in spherical coordinates:
∫dx/sinx = ln(tan(x/2))