Maxwell's equations, (charge conservation, wave equation)

Problem:

(a)  By applying charge conservation to a volume V, derive the integral form of the equation of charge conservation.  Deduce the equation of continuity, ∇∙j + (∂/∂t)ρ = 0.
(b)  Show that the equation of continuity is a consequence of Maxwell's equations.
(c)  A region containing a charge Q0 is produced at time t = 0 inside a conductor of conductivity σ.  By considering the flow of current across the surface of the region, show that at subsequent times the charge inside the region is Q(t) = Q0exp(-σt/ε0).
HINT:  Use Gauss' Law and Ohm's Law in the form j = σE.
(d)  Explain why you expect Q --> 0 as t --> ∞ .

Solution:

Problem:

(a)  Show that Maxwell's equations are consistent with the conservation of electric charge.
(b)  Show that the power P injected into a circuit by an electric field is given by ∫j∙E dV.  Verify that in steady state this reproduces the Ohmic heat loss in a “thin wire” approximation.

Solution:

Problem:

Do the fields E = i E0cos(ωt − kx ), B = 0  satisfy Maxwell's equations?
If a special condition for ρ and  j is needed, what is it?

Solution:

Problem:

(a)  Write down Maxwell's equations in vacuum for a charge density and current density free medium (ρ = 0 and j = 0).
(b)  Show that the electric field and the magnetic field satisfy a wave equation.
(c)  Write down plane-wave solutions of the wave equations for the electric field and magnetic field.  How are they related to each other?  Find the group velocity and phase velocity of the electromagnetic waves.

Solution:

Problem:

Consider an electromagnetic traveling wave with electric and magnetic fields given by
Ex = E0cos(kz – ωt) + φ),
and
By = B0cos(kz – ωt) + φ).
Using Maxwell's equations show that B0 can be written in terms of E0.

Solution:

Problem:

Use Maxwell's equations to find the magnetic field of an EM wave in vacuum for which the electric field is given by 
E
= (E0xi + E0yj)sin(ωt – kz + φ).

Solution:

Problem:

Starting with Maxwell's equations:
(a)  Derive the wave equations for a light wave in vacuum.  Write out solutions for these equations for E and B.
(b)  Show that the electric and magnetic fields are in phase, perpendicular to each other and perpendicular to the direction of motion.
(c)  Determine the relative magnitude of the E and B fields.

Solution:

Problem:

A time-dependent, vacuum electromagnetic field in three dimensions (x, y, z) at time, t = 0, is shown in the figure.

image

It has the following form:
E(r, t = 0) = i  E0exp(-(z/a)2),  B(r, t = 0) = 0.
(a) Evaluate ∂E/∂t at t = 0.
(b) Evaluate ∂B/∂t at t = 0.
(c) Evaluate ∂2E/∂t2 at t = 0 and show that E satisfies the wave equation at t = 0.
(d) What are the values of the fields E(r, t ) and B(r, t ) for a general time t, satisfying the inequality ct/a >> 1.
(e) Sketch in a single diagram the fields found in (d).

Solution:

Problem:

In unbounded free space the electric and magnetic fields satisfy

E = B = 0,  ×E = -∂B/∂t,  ×B = (1/c2)∂E/∂t,

and therefore  the homogeneous wave equation.
Assume that at t = 0 E(r, t = 0) = j f(x) and B(r, t = 0) = 0.
Find E(r,t) and B(r, t) for t > 0.

Solution: