• The electric scalar and magnetic vector potentials.
• The wave equations for the electromagnetic potentials.
A potential is a function whose derivative gives a field. Fields are associated with forces; potentials are associated with energy.
The magnetic vector potential is defined so that the magnetic field is given by:
The electric scalar potential φ is defined so that the electric field is given by:
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Note that in general, the scalar and vector potentials are functions of position and time.
The electric field in the presence of a static charge distribution is found from Coulomb’s law:
where the integral extends over all space. Note that the prime on the coordinates indicates that the coordinate is associated with the charge.
In terms of the scalar potential, for a static charge distribution, we have:
Calculating the potential is simpler than calculating the field directly; and one can then use to find the electric field.
Since we have from Maxwells’ equations:
it follows that in an homogeneous, isotropic medium:
Equation (8) is called Poisson’s equation.
Equation (4) is the solution to Poisson’s equation, expressed as an integral.
The behaviour of a charged particle in an electric field is determined by the field , rather than by the potential.
Since for an electrostatic field, we can add any function with vanishing gradient to the potential φ, and obtain the same physics. In other words, the behaviour of any electrostatic system is the same under the transformation:
where is a constant (independent of position).
The freedom that we have in choosing the potential is called gauge invariance.
This allows us to choose arbitrarily the point at which
Note that if we write the solution to Poisson’s equation (4):
then implicitly (assuming that all charges are within a finite distance from the origin), we make the gauge choice:
In the presence of sources for the magnetic field (i.e. a current distribution), the magnetic field B can be found from the Biot-Savart law:
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where the integral extends over all space.
Generally, the Biot-Savart law is difficult to apply.
It is often easier to first calculate the magnetic vector potential; but first, we need to derive the differential equation for the vector potential.
In a static case (constant fields, charges and currents), the magnetic field is related to the current density by:
Substituting , and using the vector identity:
This looks like a complicated equation; but there is a way to simplify it...
where f is some function of position. Let us define a new vector potential :
for any function the new vector potential gives exactly the same magnetic field as the old vector potential .
However, if we choose such that:
In other words, given a vector potential, we can always choose to work with another vector potential that gives the same field as the original one, but that has zero divergence.
Assuming that we make such a choice, then equation (15) for the vector potential becomes:
This is again Poisson’s equation - or rather, three Poisson equations, one for each component of the vectors involved.
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