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Chapter 12: 53 (page 568)
Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127).
The continuity equation is obtained as .
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(a) Construct a tensor (analogous to ) out of and . Use it to express Maxwell's equations inside matter in terms of the free current density .
(b) Construct the dual tensor (analogous to )
(c) Minkowski proposed the relativistic constitutive relations for linear media:
and
Where is the proper permittivity, is the proper permeability, and is the 4-velocity of the material. Show that Minkowski's formulas reproduce Eqs. 4.32 and 6.31, when the material is at rest.
(d) Work out the formulas relating D and H to E and B for a medium moving with (ordinary) velocity u.
(a) In Ex. 12.6 we found how velocities in thex direction transform when you go from Sto . Derive the analogous formulas for velocities in the y and z directions.
(b) A spotlight is mounted on a boat so that its beam makes an angle with the deck (Fig. 12.20). If this boat is then set in motion at speedv, what angle does an individual photon trajectory make with the deck, according to an observer on the dock? What angle does the beam (illuminated, say, by a light fog) make? Compare Prob. 12.10.
Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127).
(a) Write out the matrix that describes a Galilean transformation (Eq. 12.12).
(b) Write out the matrix describing a Lorentz transformation along the yaxis.
(c) Find the matrix describing a Lorentz transformation with velocity v along the x axis followed by a Lorentz transformation with velocity along they axis. Does it matter in what order the transformations are carried out?
An ideal magnetic dipole moment m is located at the origin of an inertial system that moves with speed v in the x direction with respect to inertial system S. In the vector potential is
(Eq. 5.85), and the scalar potential is zero.
(a) Find the scalar potential V in S.
(b) In the nonrelativistic limit, show that the scalar potential in S is that of an ideal electric dipole of magnitude
located at .
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