THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 401 Assignment # 4:
 
POTENTIALS, GAUGES and RELATIVITY
 
Wed. 25 Jan. 2006 - finish by Wed. 1 Feb.

Please review Section 10.1 and Ch. 12.

1. (p. 420, Problem 10.3) - GIVEN V & $\Vec{A}$ . . . Find the $\Vec{E}$, $\Vec{B}$, $\rho$ & $\Vec{J}$ corresponding to
   
   $$V(\Vec{r},t) = 0, \qquad \Vec{A}(\Vec{r},t) = {1\over4\pi\epsz} \, {qt\over r^2} \; \Hat{r} \; .$$
   

2. POINT CHARGE:
   1. Find the $\Vec{E}$ and $\Vec{B}$ fields corresponding to a stationary point charge q situated at the origin.
   2. State the charge and current distributions of this situation.
   3. What are the electric and magnetic potentials?
   4. Is there any relation between this situation and that described in Problem 10.3?

3. (p. 420, Problem 10.5) - GAUGE TRANSFORMATION: Use the gauge function
   
   $$\lambda = -{1\over4\pi\epsz} \; \left(q \, t\over r\right)$$
   
   
   to transform the potentials in Problem 10.3, and comment on the result.

4. WHICH GAUGE?
   1. In Problem 10.3 above, are the potentials in the Coulomb gauge, the Lorentz gauge, both, or neither?
   2. In Problem 2 above, are the potentials in the Coulomb gauge, the Lorentz gauge, both, or neither?

5. NATURAL UNITS: Since c is now a defined quantity that keeps appearing in confusing places in our notation for 4-vectors etc., and since nanoseconds (ns) are perfectly handy units for distance, it seems silly to not just measure time and distance in the same units (seconds) and set c=1. While we're at it, why not set the ubiquitous constant in quantum mechanics to unity as well ($\hbar = 1$) so that all angular momenta are unitless and (because $E = \hbar \omega$) energies are measured in s^-1?
   1. In what units would we then measure velocities, momenta, masses, forces and accelerations?
   2. Suppose we set the Coulomb force constant $k_E \equiv \kE = 1$ as well. In what units would we then measure charge, electric field, magnetic field, and potentials V and $\Vec{A}$?
   3. Write out Maxwell's equations in this system of units. (Hint: We must have $\epsz \, \muz = 1$.)¹

6. 4-POTENTIAL: In Eq. (12.131) on p. 541, Griffiths states that, "As you might guess, V and $\Vec{A}$ together constitute a 4-vector: $A^\mu = (V/c, A_x, A_y, A_z)$." This is a very strong statement with profound consequences. You can't just take any 3-vector and combine it with a convenient scalar in the same units to make a true 4-vector! Explain why we should believe this about $A^\mu$, and list any essential conditions that must be met for it to be true.