A thin ring of radius R made from a dielectric carries a total charge Q
uniformly distributed over its circumference. It lies in the xy-plane
centered at the origin and is rotating about the z-axis with angular velocity
(a) Write down a lowest-order expression for E(r) for r >> R.
(b) Write down a lowest-order expression for B(r) for r >> R.
(c) Does the ring produce electromagnetic radiation? Why or why not?
(d) Is the Pointing vector zero or nonzero for r >> R?
A finite spherically symmetrical charge distribution disperses under
the influence of mutually repulsive forces.
Suppose that the charge density ρ(r, t), as a function of the distance r from the center of symmetry and of time, is known.
(a) Prove that the curl of the magnetic field is zero at any point.
(b) Use this to show that B is zero at any point.
Assume magnetic charges exist and Maxwell's equations are of the form
∇∙E = ρ/ε0, ∇∙B = μ0ρm, -∇×E = μ0jm + ∂B/∂t, ∇×B = μ0j + μ0ε0∂E/∂t.
Assume a magnetic monopole of magnetic charge qm is located at the origin, and an electric charge qe is placed on the z-axis at a distance R from it.
(a) Write down expressions for the electric field E(r) and the magnetic field B(r). Make a sketch.
(b) Write down expressions for the momentum density g(r) and angular momentum density ℒ(r) of the electromagnetic field.
(c) Show that there is electromagnetic angular momentum Lz about the z-axis and derive an expression for it. Show that Lz is independent of R.
Useful vector identity: (a∙∇)n = (1/r)[a - n(a∙n)]. Here n is r/r is the unit radial vector.
Given are Maxwell's equations in macroscopic
∇∙D = ρf, ∇×E = -∂B/∂t, ∇∙B = 0, ∇×H = jf + ∂D/∂t,
and the constitutive relations for a linear, isotropic medium
D = εE, B = μH.
Let jf = σcE + jb, where σc is the conductivity of the medium and jb represents the density of forced currents, such as currents inside a battery.
(a) What is the relationship between the field quantities E and
and the potentials A and V?
(b) E and B do not specify A and V uniquely. Let A = A' - ∇ψ, V = V' + ∂ψ/∂t. Show that Maxwell's equations are satisfied by A' and V' if they are satisfied by A and V.
(c) Briefly discuss the gauge transformation represented in part b and its implications.
(d) Write the relationship between the fields given in answers to part (a) in terms of H, D, A, and V.
(e) Find the wave equation for A in terms of V, μ , ε , jb, σc , and v = (μ0ε0)-½ in the Lorentz gauge.
(f) Starting with Maxwell's equations, derive the wave equation for V in the Lorentz gauge.
A thin wire of radius b is used to form a circular wire loop of radius a (a
>> b) and total resistance R.
The loop is rotating about the z-axis with constant angular velocity ωk in a region with constant magnetic field B = B0i.
At t = 0 the loop lies in the y-z plane and the point A at the center of the wire crosses the y-axis.
Let the (θ/θ) direction be tangential to the loop and be equal to the positive z direction at point A.
Let the (φ/φ) direction be tangential to the wire and be equal to the direction indicated in the figure.
(a) Find the current flowing in the loop. Neglect the self-inductance of the loop. What is current density J as a function of time?
(b) Find the thermal energy generated per unit time, averaged over one revolution.
(c) Write down an expression for the the pointing vector S on the surface of the wire.
(d) Use S to find the field energy per unit time flowing into the wire, averaged over one revolution.