March 21, 2007
PHY 712 – Problem Set # 18
Continue reading Chapter 6 of Jackson.
1. In the Lorentz gauge and in the absence of sources, we found that the vector A(r, t)
and scalar Φ(r, t) potentials must satisfy the following equations:
∇2Φ(r, t)−1
c2
∂2
∂t2Φ(r, t) = 0 (1)
and
∇2A(r, t)−1
c2
∂2
∂t2A(r, t) = 0.(2)
Assuming solutions of the form:
Φ(r, t)≡Φ0ei(k·r−ωt)A(r, t)≡A0ei(k·r−ωt),(3)
where Φ0and ωare scalar constants and Aand kare vector constants, find relation-
ships between these constants that must be satisfied in order to satisfy the Maxwell’s
equations and the Lorentz gauge conditions. Also, determine the corresponding forms
of the Eand Bfields.