Maxwell's Equations and the Vector Potential.


The original version of Maxwell's equations was a set of 20 equations, but some are just sets of three in the three coordinates, which in vector notation reduces the total to 8. One is about charge conservation and another is Ohm's Law, but there is also an extra field not included in the modern set of 4 field equations, the 'vector potential' (A). Being nominally from a Physics background I had to learn about A, and there is good reason to believe that it is in some sense more fundamental than the usual electric (E) and magnetic (B) fields. There are experiments demonstrating the Aharonov-Bohm effect where the E and B fields are arbitrarily small, but a non-zero vector potential has a significant effect on electron diffraction, and also in the theory of quantum electrodynamics the vector potential plays a central roll. So why is it widely ignored in Electronics? When we talk about 'voltage' this is just the 'scalar potential' (and also we could ask: if it is a quantum mechanical effect how did Maxwell know anything about it back in 1864?)

The original equations are not very useful or interesting, so let's jump straight to the more modern set of 4 equations now known as 'Maxwell's Equations'. The inverted triangle symbol is the 'vector differential operator' sometimes called 'del' or 'nabla'. We don't need to go into all the details of what these equations mean, and in any case it is far easier to understand them in a different form using 'flux' and 'circulation', which I will get to eventually.

E, B and J are in bold letters to indicate they are vectors. (i.e. they have both magnitude and direction.)
E and B can be defined by a single equation for the force on a charge q.

E determines the force qE on a stationary charge and B determines the additional force q(v x B) when the charge moves at velocity v.
v x B is the 'cross product' of the two vectors, and if both vectors are in the horizontal plane as in the next example, with angle 'a' from v to B then the force q(v x B) is vertically upwards and has magnitude qvB sin(a).

Different observers moving at different velocities will disagree about the velocity of the charge. They may also find different values for the electric and magnetic fields, but the relationship between the charges and fields will still be correctly given by Maxwell's equations. In other words the equations are 'relativistically invariant'. So, Maxwell appears to have produced equations including some quantum mechanical effects and conforming to the principles of Special Relativity back in 1864 before either theory was known.

When Maxwell introduced the vector potential of course he knew nothing about quantum mechanics, it was only about field momentum. This had previously been considered by Faraday, and Maxwell's idea was that near an electric charge q the electromagnetic momentum is given by qA/c where c is the speed of light. This new field A is related to the magnetic field B by the following equation:

For a given magnetic field B the A field is not uniquely defined by this relationship, it is necessary to add what is known as a 'gauge condition', such as the 'Coulomb-gauge' which specifies the divergence of A to be zero. There are other gauges such as the 'Lorenz gauge', but they have no observable effect, they are generally chosen for mathematical convenience. This lack of effect is called 'gauge invariance'. The only feature having any observable effect is the line-integral of A round a closed loop. There is also an arbitrary feature of the scalar potential, we can only observe potential differences between two locations, and in practice we usually choose some convenient location to define as zero voltage and measure all other voltages relative to that point.

So why do we call this field the 'vector potential'? The more familiar 'scalar potential' is what we call voltage (V), or potential difference. The electric field along a line, e.g. in the x direction, is then given by: E = -dV/dx. For a uniform field it is just voltage divided by distance, (E has units of volts/metre).
A more complete equation for the electric field however includes the vector potential, and is:

The first term is the gradient of the scalar potential (voltage) in 3 dimensional space. The second term in this equation is zero unless there is a varying magnetic field, and then, circling back to equation 3 above we find another way to express the connection between the electric field and a varying magnetic field. Equation 3 is commonly known as 'Faraday's Law', and the effect of a varying magnetic field through a loop is then said to cause an emf (electromotive force) round the loop. So, we can describe the same effect as either an emf via equation 3, or as a vector potential (A) via equation 7. So, by using emf instead we can get by just fine without needing to understand the vector potential, unless we encounter the quantum mechanical effects mentioned above.

Equations 1 to 4 are Maxwell's equations in what we could call 'differential form', on the grounds that they use the differential operator. They are however often easier to apply in 'integral form'. We then use 'flux', or 'surface integral', to specify the average field component perpendicular to a surface multiplied by the surface area, and 'circulation', or 'line integral', to specify the average field component along a loop multiplied by the distance round the loop. The 4 equations then become:

We can choose either direction through a surface or a loop to define as 'positive', but there are then two possible directions to choose as the positive direction round the loop. This is just a sign convention, the one chosen (I don't know who got to choose, possibly Fleming, who is credited with some similar conventions) is a 'right-hand rule', if we point in the positive direction through the loop with the thumb of our right hand and curl our fingers round then they point in the positive direction round the loop. The positive directions are shown by arrows in the next diagram:

Both equations 10 and 11 include line integrals round a loop, and if we had used the opposite convention for the positive direction round the loop then instead of a minus sign in eq.10 it would appear in eq.11, applied to both terms on the right hand side. That minus sign causes some confusion regarding the interpretation of eq.10, also known as 'Faraday's Law'. This has somehow evolved into 'Lenz's Law' which states that a varying magnetic field through a conducting loop will cause a current round the loop which causes an additional magnetic field which 'opposes' the original change in the field. There are situations in which that is a reasonable interpretation, and others where it appears to be confusing, but it is not an inevitable consequence of that minus sign, which is a result of this sign convention.

As an example of how A and E are related to the magnetic field B imagine a magnetic field confined to a central circular region and directed perpendicular into the page, as signified by the '+' signs in the next diagram. This could be achieved by a long solenoid; it needs to be long to keep the external field small, so we only need to consider the central magnetic field and whatever that causes externally.

If B is constant as in the first diagram then equations 3 and 10 both tell us that the electric field round a loop is a function of the rate of change of B which in this case is zero, so E is also zero. Equation 6 however tells us the vector potential is determined by the value of B, not its rate of change, so there is a non-zero vector potential. If we convert equation 6 to integral form we find the line integral of A round the loop is equal to the 'flux' of B through the loop. If the inner circle containing the (uniform) magnetic field has area a then the flux of B is just aB, and if the outer loop round which we want to determine A has circumference c then the line integral of A round the loop is cA. Then aB = cA, where B and A are the magnitudes of B and A. Because the fields are uniform the flux of B is just the field through a surface multiplied by its area, and the circulation of A is just the field along the loop multiplied by the distance round the loop.

If B is varying as in the second diagram then there will be a circulating E field. The minus sign in equation 3 or 10 tells us which direction the electric field goes round the loop. The direction shown in the diagram is correct if the magnetic field shown is increasing in magnitude. There is still a circulating A field, in this case in opposite direction to the electric field, and varying in proportion to the magnetic field.

The 'loops' so far have just been circular paths in space, but we could add a conducting loop to see how this is affected. We need to be clear about the properties of this conductor, assume it to be a wire with significant resistance, which we can draw as a series of ten 1R resistors.

The current round the loop will generate its own magnetic field and may need taking into account if the loop consequently has significant inductance, but for this example we assume the resistance is high enough to be the dominant effect for the rate of change of B we wish to consider.

Again assume an increasing magnetic field confined to a central region, directed into the page. As previously there will be an electric field in an anti-clockwise direction round the loop, which will now cause a current to flow round the conductor. Suppose the current is 1 amp, then applying Ohm's Law we would expect a voltage of 1V across each resistor, and for these to add as we progress round the loop. Define the starting point as zero volts, then when we get back to the starting point we expect an extra 10V, but we are back to the starting point and the voltage there is still zero, so we are forced to conclude that there must be zero voltage across each 1 ohm resistor carrying current 1 amp. So we find just applying Ohm's Law gives us the wrong answer. What we need is to go back to equation 7 which tells us that the field E = -dV/dx -dA/dt. The scalar potential is zero everywhere round the loop, but there is still an electric field causing a current to flow because of the varying vector potential.

Having confined the magnetic fields to small areas inside the loops gives the impression of 'action at a distance'. For a varying magnetic field there will be radiated electromagnetic energy which can transfer energy to the loop, but for a constant field the vector potential is still there at any finite distance from the central region. That does look like action at a distance, but a constant vector potential has no associated energy, and is not directly observable. It can have a quantum mechanical effect, the phase of an electron wave function can be changed by the field, but that also is not directly observable, only the differences in phase along two possible electron paths can affect the resulting diffraction patterns. These two paths form a loop, and it is a non-zero line integral of A round the loop which causes different phase shifts along the two paths and changes to the electron diffraction pattern.

The example with a resistive loop has a few subtleties worth a mention. Even assuming everything is electrically neutral and the rate of change of B is constant, giving a constant current, neither the charge density nor the current density are uniform across a cross-section of the conductor, for a number of reasons (e.g. think of centripetal force acting on the conduction electrons following a circular path). My favourite explanation of current distribution was by Feynman, that the distribution is the one giving the minimum increase in entropy. That is so simple to say, but in practice almost entirely unhelpful, and possibly wrong. See Feynman Vol-2 Section 19-2.
More prosaically we could note that the distance round the loop is shorter round the inside surface compared to the outside, so has lower total loop resistance, and the electric field is also stronger, so if that is the dominant effect the current will be higher on the inside.


Footnote:
Having said a few unkind words about 'Lenz's Law' it's the turn of Kirchoffs Second Law. There are a few different versions, not all wrong, but one example I found is:
'The net electromotive force (emf) around a closed circuit loop is equal to the sum of potential drops around the loop.'
In the previous conducting loop diagram above the line integral of the electric field round the loop is what is referred to as the emf, but with zero potential drops round the loop. That version of Kirchoff's Law must be wrong, there is a non-zero emf together with zero potential drops. What is missing is the condition that it applies only to 'conservative fields' which is not true for induced electric fields. What is a 'conservative field'? It's a field in which a charge can follow any path and return to its starting point with zero net energy gain or loss from the field. Going round a loop with a circulating electric field a charge will gain or lose energy depending on its direction, so that field is non-conservative.

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