potential and fields

POTENTIAL AND FIELDSBy:DR. SAKTIOTO, M.PHIL

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)


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)


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)

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)

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)

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potential and fields