Maxwell's Equations

• Problem:
  Use Maxwell's equations in SI units.

  (a)  Show that the electric field in free space obeys the wave equation.

  (b)  Show that the propagation velocity of the wave is given by .

  (c)  Show that the time-varying electric field is perpendicular to the velocity of propagation.

  (d)  Show that the time varying magnetic field is perpendicular to both E and the direction of propagation and that .

  In parts c and d you may assume plane waves for simplicity.

  • Solution:

    

• Problem:

  (a)  Beginning with Maxwell’s equations, show that in the Lorentz gauge the vector potential A satisfies the inhomogeneous wave equation.

  (b)  Assume that the source and the fields have a time dependence Show that he vector potential obeys the inhomogeneous Helmholz equation. Write the solution of this equation in terms of Green’s functions. Make the dipole approximation and show that in the radiation zone the vector potential is is the electric dipole moment of the source.

  (c)  How would you calculate the total power emitted by this source. A qualitative answer without detailed calculations will be sufficient.

  • Solution:

    

    

• Problem:
  A current sheet of infinite extend, located in the plane x=b, is established parallel with an infinite, ideal, plane conductor at x=0.  The current density is given by

  ,

  where d is the Dirac delta function.  The medium in the region x>0 has a dielectric permittivity e and a magnetic permeability m .

  

  (a)  Show that the vector potential may be written as , where .
  HINT: The Green's function for the differential equation is .

  (b)  Write e=e₀(1+ia), where a is a positive infinitesimally small damping constant. Show that when , the electric field may be written , where .  Discuss the interpretation of this equation.

  • Solution:

    

    

• Problem:
  Show that a radially oscillating spherically symmetric charge distribution does not radiate.  Assume a harmonic time variation of frequency w and use the complex formalism, namely , and so forth.

  (a)  First write down Maxwell's equations and the continuity equation using the complex formalism to eliminate time derivatives.
  (b)  Note that one can write J(r)=rf(r). Compare Ñ×D and Ñ×J to relate D and J.
  (c)  Show that
  (d)  What are D and B outside the source? (Hint: Consider Ñ×B and Ñ´B.)
  (e)  What do you conclude concerning radial oscillations of a spherically symmetric source with an arbitrary time variation?

  • Solution:

    

• Problem:
  A carbon dioxide molecule has a linear configuration O-C-O. Assume that each oxygen atom has a charge +q and that the carbon has a charge -2q. Further assume that the constant C-O distance is d and that the molecule rotates with constant angular speed w about an axis perpendicular to its length.

  (a)  Write the electric monopole, dipole, and quadrupole moments of the molecule relative to coordinates fixed in the molecule.
  (b)  For speeds small compared to the speed of light, find the scalar and vector potentials at a point P in the plane of rotation at a distance R>>d , excluding terms of order (d/R)² and higher.
  (c)  Find the electric and magnetic fields at P, excluding terms of order (d/R)² and higher.

  • Solution:

    

    

• Problem:

  Consider a plasma made up of electric charges of mass m and charge e.  Suppose that the density of particles, n, is uniform.  Assume also that the interaction between the charges may be neglected (i.e., the damping force is zero).  Electromagnetic plane waves of frequency w and wave vector k are incident on the plasma.
  (a)  Obtain the conductivity, s, entering Ohm’s law j=sE as a function of frequency.  Hint: Use Newton’s equation of motion for the charges in the presence of the electric field.
  (b)  Obtain the dispersion relation - the relation between the magnitude of the wave vector k and the frequency w - for the solution of Maxwell’s equations in the plasma whose conductivity you obtained in part (a).
  (c)  Suppose now that there is an external uniform, static magnetic field B₀.  Consider plane waves traveling parallel to B₀.  Show that the index of refraction, n=ck/w is different for right and left circularly polarized waves.  For simplicity, assume that the magnetic field of the incident wave is small compared to |B₀|.  Hint: Recall that the polarization vector for right (left) circularly polarized waves is given by

  ,

  where e_x and e_y are unit vectors, and we have assumed that B₀ is directed along the z-axis.

  • Solution: