1. From your electrodynamics course, you should recall that the voltage (the scalar po- tential) of a stationary point charge q a distance r away is simply V(r) 1 q 4περτ (1) where can be written in terms of Cartesian coordinates as r = √√x² + y² + z². Fur- thermore, for a static charge, there is no vector potential, A (r) = 0. The electric and magnetic fields can then be found from Ē = B = -VV - A V × A. Ət (2) In the static case, the electric field simply gives back Coulomb's law, while the magnetic field vanishes, as you know. One topic that you probably didn't work out, however, is what are the potentials of a charge that is moving? If the charges are moving slowly enough (compared to light), then the scalar and vector potentials reduce to the above forms, but what if they are moving faster so we have to worry about the fact that signals can't travel faster than light? That's what we want to investigate using Relativity. Here's how we're going to do it. Because the scalar and vector potentials together form a four vector, Aμ = - (ˇ,A), (3) where c is the speed of light, we know how the components transform under a Lorentz boost. (a) Suppose that Alice is looking running past Bob at a velocity v, and holding a static charge q. She sees a static charge so that her scalar potential will be V (r) = = 1 q 4πЄor' while her vector potential will be 1 = A' (r) = 0. զ (4) (5) Using the fact that the scalar and vector potentials transform as a four-vector, show that Bob will see a scalar potential 1 γα (6) where = potential V (r) = Απερ √72 (2 √ √² (x − vt)² + y² + z² 1/√1 v2/c². In addition, show that Bob will also see a vector v A (r) = V (r) c2 (7) where is the velocity in the x-direcdtion, and with the scalar potential given in Eq. (6). (b) Show that, for the velocity in the x-direction, Eq. (6) can be written q 1 V (r) Απεργ (1 v.) - (8) Hint: Note that the time, t, showing up in Eq. (6) is the time it takes for the information about the potential to reach a distance r, so t = r/c. Then, write down the vector potential using Eqs. (7) and (8). These potentials are called the Liénard-Wiechert Potentials, and turn out to be true for a charge moving in any way.