Question

1 9 Consider a point charge at rest at the origin. You know that V(F,t) a. Explain why, in this case, A(F,t) 0. b. Explicitly check to see if we are in the Coulomb gauge, the Lorenz gauge, or perhaps c. Introduce a gauge transformation AnewA Vf, Vnew - V - ôf/ôt, using the d. Briefly discuss your results: are the potentials "static" or time-dependent? Does this both or neithei particular choice f(r, t)-qt/4?6r. Find Anew and hew represent the same physical situation, or is something different here? Are we in the Coulomb gauge or Lorenz gauge now? Do we have to be in one of those to solve physics problems? What has changed, and what is the same? The Coulomb Gauge. As in magnetostatics, we pick The Lorenz gauge. In the Lorenz2 gauge, we pick av Ot

Answer 1

(a) V (r, t) = 1/4 pi epsilon_0 a/r And for a point charge at rest, the charge density is rho (r, t) = q delta (r) and v = 0 so, the current density, J = rho v = 0 And as a point charge at rest does notequalto create any magnetic field, so B = O So, from nabla times A = B, we get, A = O (b) nabla middot A + mu_0 epsilon_0 Partial differential V/Partial differential t = 0 + 0 = 0 and nabla middot A = 0 Thus the two are satisfied (c) A_new = A + nabla f, V_new = V - Partial differential f/Partial differential t where, f = q t/4 pi epsilon_0 r implies A_new = A + nabla f = 0 + (- qt/4 pi epsilon_0 r^2 r) implies A + new = V - Partial differential f/Partial differential t = q/4 pi epsilon_0 r - q/4 pi epsilon_0 r = 0 implies V_new = 0 (d) A_new = -qt/4 pi epsilon_0 (r/r^2) is time dependent and V_new = 0 Thus