Fall 2012 Homework Assignments
1) Consider the function f(x) = -|x| . Calculate the quantities E(x) = -df/dx and V(x) = -d2f/dx2 . Integrate V(x) near the origin as ∫-εε V(x) dx by parts, where ε is a small quantity. Show that the result is finite. 2) Consider the function f(r) = - ln(r) where r2 = x2 + y2 . Calculate the quantities E = -∇f and V = -∇2f where ∇ = i ∂/∂x + j ∂/∂y and i and j are the unit vectors in the x and y directions respectively. Integrate V near the origin as ∫-εε dy ∫-εε dx V(x,y) by parts, where ε is a small quantity. Show that the result is finite.
1) Consider four point charges located at the four corners of a square with sides of length a. The amount of charge and their coordinates are as follows: charge x-coord y-coord Charge 1 : +q a/2 a/2 Charge 2 : -q a/2 -a/2 Charge 3 : +q -a/2 -a/2 Charge 4 : -q -a/2 a/2 Note that the total charge is zero and the charge distribution may be viewed as two dipoles in opposite directions. Such charge densities are called quadripoles. (You can set the "quadripole moment" Q = q a2 .) Find the electrostatic potential at large distances r from the origin. (i.e. r >> a). Comment on special cases for the potential on the x, y, and z axes.
1) A conducting sphere of radius a and at zero potential is placed at the origin. A line charge with charge density (charge per unit length) λ is placed on the positive z axis between the points z=2a and z=3a. o Find the image charge density inside the sphere in order to have zero potential on the spherical surface. (Note that the image charge density is not uniform.) o Find an expression for the potential for any point outside the sphere. o Work out the integral in the previous part to determine the potential on the negative z axis for |z| ≥ a. Evaluate and discuss the limits as |z| → a and |z| → ∞
1) A conducting cubic box has all faces at zero potential. Side of the box has length a. The box is filled with a total charge Q distributed unifomly inside the box. Find the potential at points inside the box. 2) What is the total electrostatic energy associated with the above charge density?
1) The region indicated in the figure has cylindrical symmetry and extends to infinity in the ±z directions. The boundary of the region at radius a is kept at potential -V0 and that at radius b is kept potential V0. Other boundaries are kept at zero potential. Find the potential inside the region. The next two parts are for extra credit: 2) For the region indicated above, find the Green's function corresponding to a unit line charge (extending in the z direction) placed at the position ρ' , φ' inside the reqion, when all boundaries of the region are kept at zero potential. 3) The region is now filled with a charge density μ (charge per unit volume). Find the energy stored in the electrostatic field inside the region, within a volume which extends a length L in the z direction.
1) A ring-charge of total magnitude Q and radius a is placed inside a grounded sphere (V=0) of radius R. (The charge is uniformly distributed on the ring.) The system has azimuthal symmetry and all point of the ring are a distance b from the center of the sphere as shown in the figure. Find the potential in terms of the spherical eigenfunctions at all points inside the sphere. Find the potential outside the grounded sphere when the ring is also outside the sphere (i.e. when b > R).
1) Half of the space for z < 0 is filled with a dielectric with permittivity ε. Rest of the space is vacuum with permittivity ε0. A dipole P is placed a distance D from the origin, on the positive z-axis. The dipole points in the +z direction. Find the potential at all points in space.
1) Half of the space for z < 0 is filled with a magnetic material with permeability μ. Rest of the space is vacuum with permeability μ0. A wire parallel to the x axis and a distance D from from the x-y plane carries a current I in the +x direction. Use the result of problem 5.17 in Jackson to find the magnetic field B at all points in space. Make a sketch of the field lines in the y-z plane for the cases μ=μ0, μ=2μ0, and μ>>μ0.
1) A circular ring of current I with radius a is placed in a uniform magnetic field B0 in the +z direction. The field makes an angle θ with a normal to the plane of the ring. Find the exact value of the torque on the ring.
1) Problem 5.23 in Jackson.
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