We want general solution to Maxwell's Equations,
Given what are the fields
From equation (ii) we can still write
Now from equation (i)
From equation (iv)
Equations (1) and (2) contain all the information of Maxwell’s equations. Thus we need to calculate only four components (one for V and three for instead of six components (three for and three for
Example 1: In an infinite straight wire constant current I0 is turned on abruptly at t = 0 . Then corresponding retarded potentials are given by . Find the fields corresponding to these potentials.
The electric field is
And the magnetic field is
Suppose we have two sets of potentials, which correspond to the same electric and magnetic fields.
Two potentials also give the same
Actually, we might as well absorb k (t ) into λ, defining a new λ by adding to the old one. This will not affect the gradient of λ ; it just adds
It follows that
Conclusion: For any old scalar function λ , we can add
provided we simultaneously subtract from V. None of these will affect the physical quantities and Such changes in V and are called gauge transformations.
Coulomb Gauge reads
Lorentz Gauge condition is
Using Lorentz Gauge condition
The virtue of the Lorentz gauge is that it treats V and on an equal footing: the same
differential operator (called the d' Alembertian) occurs in both equations:
Example 2: For a vector potential the divergence of is where Q0 is a constant of appropriate dimensions. Find the corresponding scalar potential that makes and V Lorentz gauge invariant.