The document discusses Maxwell's equations and how they can be formulated using electric and magnetic potentials. It shows that Maxwell's equations are automatically satisfied if the electric and magnetic fields are written in terms of scalar and vector potentials. It derives the Lorentz gauge condition and shows that under this condition, Maxwell's equations reduce to a simpler form involving only the vector potential. It then discusses how Maxwell's equations can be solved using retarded potentials and Green's functions, with the retarded potentials depending on the retarded time, which is the earliest time at which a signal could travel from the source to the observation point.

1. POTENTIAL AND FIELDS
By:
DR. SAKTIOTO, M.PHIL

2. Potential Formulation
We seek the general solution to Maxwell’s equations:
We have seen that Eqs. (10.2) and (10.3) are automatically satisfied if we
write the electric and magnetic fields in terms of potentials
(10.1)
(10.2)
(10.3)
(10.4)
(10.5)
(10.6)

3. Potential Formulation
This prescription is not unique, but we can make it unique by adopting the
following conventions:
The above equations can be combined with Eq. (10.1) to give:
Let us now consider Eq. (10.4). Substitution of Eqs. (10.5) and (10.6) into
this formula yields
or
(10.7)
(10.8)
(10.9)
(10.10)
(10.11)

4. Potential Formulation
We can now see quite clearly where the Lorentz gauge condition comes
from. The above equation is, in general, very complicated, since it
involves both the vector and scalar potentials. But, if we adopt the Lorentz
gauge, then the last term on the right-hand side becomes zero, and the
equation simplifies considerably, such that it only involves the vector
potential. Thus, we find that Maxwell's equations reduce to the following:
This is the same (scalar) equation written four times over. In steady-state
(i.e. , ), it reduces to Poisson's equation, which we know how to
solve. With the terms included, it becomes a slightly more
complicated equation (in fact, a driven three-dimensional wave equation).
(10.12)
(10.13)

5. Retarded Potentials
We are now in a position to solve Maxwell's equations. Recall that in steady-
state, Maxwell's equations reduce to :
The solutions to these equations are easily found using the Green's function
for Poisson's equation:
We can solve these equations using the time-dependent Green's function :
(10.14)
(10.15)
(10.16)
(10.17)
(10.18)

6. Retarded Potentials
with a similar equation for . Using the well-known property of delta-
functions, these equations reduce to
What is this earlier time? It is simply the latest time at which a light signal
emitted from position r’ would be received at position r before time t .
This is called the retarded time. Likewise, the potentials (10.19) and
(10.20) are called retarded potentials. It is often useful to adopt the
following notation
(10.19)
(10.20)
(10.21)