Question

Problem 3 Consider a possible solution to Maxwell's equations in vacuum given by A(x, t) = Ao exp(i(kx - wt)), V(x, t) = 0 where A is the vector potential and V is the scalar potential. Suppose Ao, k and w are constants in space and time. a) Compute the time-dependent electric and magnetic fields from the given potentials. Show your work. b) Give the contraints, if any, on Ao, k and w imposed by the following two Maxwell's equations in vacuum (show your work): (i) V.E=0 (ii) DXE - OB Give a physical interpretation of each of the constraints. (If you could not solve part a), then use expression for a planar electromagnetic wave propagating in y-direction in part b).).

Answer 1

gines An, t) = Ao exp Ci(kr-wt)) vnit) - O a) Electric field E=JV-DA at Tv=o as veto=0 A = Ao explíckn-wt) iw =-A swiexplickr-wot)) E =-(-Agico exp (ickn-wot) = Airo exp Cike-wt)) magnetic field B=JXÃ = JX Ao exp (i knowt)

Let Ão = Aron ^ + Aloy 9 + Ang 3 explickn-wot)) Aon Aoy Aog my Aon explicken-wit) Any explick ge-est) Aoz explickr-wts = ^ (Asy explicka-est) ik xo) –} (Aoz ik expléter-wt)) ľ (hoy ik (explickr-wt) B = ikhoz exp (i (kn-est)) } t ik Aoy explickr-wt)) b) v = D.Ciw A, explicken-wt)) =iw ( non it-exp Cillen-cut) =-kw Hon exp ik n-w ) has to be zero

Je =0 it and only if for o Physically which means that Ē & the electric field cannot have any component on a direction on the direction of propagation. In other words ' has to be transure iw / de دلية - Aon explickr wt) Aoy expliler-wt) Az explicanot) =-kw (Az explicken wit )) 9 + Hoy exp Ciken-wt))}) OB = ik(Aoz explican-wt))-iwift Aoy explicar-we) = kw (Aoz exp Cickn- wt) ŷ thog explickn-wt) OXĚ = -2B ploned P.S: Please do nate the answer be