Spring 2012 Homework Assignments
Consider a dipole made up of a charge +q at the (x,y,z) coordinate (0,0,a/2) and a charge -q at (0,0, -a/2). The dipole moment is then p = aq. Find the electric field at a point (x,0,z) far away from the origin. You should consider the limit when r >> a where r^{2} = x^{2} + z^{2}. Express your result in terms of r and θ, where θ is the angle between the x-axis and the position at (x,0,z): tan(θ) = z/x so that x = r cos(θ) and z = r sin(θ).
A spherically symmetric charge of radius R has a charge density (per unit volume) equal to ρ(r). (a) What must ρ(r) be so that the field inside ( r < R ) has a constant magnitude E_{0} ? (b) What is the total charge in a volume of radius r ? (c) Plot the quantities in parts (a) and (b) for all values of r. Repeat the parts (a), (b), and (c) above for an infinitely long cylindrical charge of radius R.
Consider the two geometries in Homework 2. In both cases, assume that the potential at r = 0 is equal to zero. For both cases, find the potential as a function of r, and plot your results.
Find the magnetic field on the axis of a rotating thin charged disk. The insulating disk is uniformly charged with a surface charge density (per unit area) σ and has a radius R. The disk is rotating with angular velocity ω.
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