PHY 415: Electromagnetic Theory I
Prof. S. Teitel firstname.lastname@example.org ---- Fall 2007
Problem Set 3
Due Monday, October 22, in lecture
- Problem 1 [10 points total]
Consider an infinitely long grounded metal cylinder, of radius R, placed at right angles to an applied uniform electric field Eo.
a) Find the potential outside the cylinder. [5 pts]
b) Find the surface charge density σ induced on the surface of the cylinder. [5 pts]
- Problem 2 [30 points]
a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σo, that is lying in the xy plane centered on the orign. Using the Coulomb formula for the potential,
φ(r) = ∫ da´
|r - r´|
do the integral to find the exact value of the potential φ for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the following parts!). [5 pts]
b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential φ(r) in terms of a Legendre polynomial series using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order l=5, for |r|>R and |r|<R, both above and below the disk. (Hint: read Jackson section 3.3) [10 pts]
c) Using your results of part (b), compute the electric field E just above and just below the disk, and explicitly show that the discontinuity in E is given by the surface charge density σo, as we know from general principles that it must be. [5 pts]
d) Now, keeping the origin at the center of the disk, compute the monopole moment, the dipole moment vector, and the quadrapole moment tensor of the charged disk. Use these moments to write an approximation for the potenial φ(r) far from the disk (i.e. r >> R). Compare with your results from part (b). [10 pts]
- Problem 3 [10 points]
Given the quadrapole tensor Q in the coordinate system r, derive an expression for the quadrapole tensor Q´ in the coordinate system r´, where r´ = r - d (d is a constant displacement vector). Show that Q = Q´ (i.e. that the quadrapole moment is independent of the choice of orign) only if both the monopole and the dipole moments vanish.
- Problem 4 [10 points]
Consider a charge +q lying at position +s/2 on the z axis, and a charge -q lying at position -s/2 on the z axis.
a) Using Coulomb's law, write down an exact expression for the potential φ(r) at any point r. Express your answer in spherical coordinates. [2 pts]
b) When s<<r, one can approximate this exact expression by a power series expansion in (s/r). Derive this expansion directly by doing a Taylor series expansion of your answer for part (a). Carry out the expansion to order (s/r)3. [4 pts]
c) By rewriting your answer for part (b) in terms of a series of Legendre polynomials, identify the dipole, quadrupole, and octopole terms in this expansion. [4 pts]
- Problem 5 [15 points]
a) Consider a spherical shell of radius R, with uniform surface charge density σo, centered on the origin. The shell is spining counterclockwise about the z axis with angular velocity ω. Find the magnetic vector potential A(r), far from the sphere, using the magnetic dipole approximation. Find the magnetic field B within this approximation [7 pts].
b) Using the method of separation of variables, as applied to the scalar magnetic potential φM, find an expression for the exact magnetic field B both inside and outside the spining charged shell of part (a). How does your answer for the field outside compare with that obtained by the magnetic dipole approximation in part (a)? [8 pts]