*Michael Fowler, Physics
Department, UVa 5/9/09*

Maxwell’s four equations describe the electric and magnetic
fields arising from varying distributions of electric charges and currents, and
how those fields change in time. The
equations were the mathematical distillation of decades of experimental
observations of the electric and magnetic effects of charges and currents. Maxwell’s own contribution is just the last term
of the last equationbut
realizing the necessity of that term had dramatic consequences. It made evident for the first time that
varying electric and magnetic fields could feed off each otherthese
fields could propagate indefinitely through space, far from the varying charges
and currents where they originated.
Previously the fields had been envisioned as tethered to the charges and
currents giving rise to them. Maxwell’s
new term (he called it the *displacement
current*) freed them to move through space in a self-sustaining fashion, and
even predicted their velocityit
was the velocity of light!

Here are the equations:

- Gauss’ Law for electric fields: (The integral of the outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units.)
- The corresponding formula for magnetic fields: (No magnetic charge exists: no “monopoles”.)
- Faraday’s Law of Magnetic Induction: The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit.
- Ampere’s Law plus Maxwell’s displacement current: This gives the total magnetic force around a circuit in terms of the current through the circuit, plus any varying electric field through the circuit (that’s the “displacement current”).

The purpose of this lecture is to review the first three equations and the original Ampere’s law fairly briefly, as they were already covered earlier in the course, then to demonstrate why the displacement current term must be added for consistency, and finally to show, without using differential equations, how measured values of static electrical and magnetic attraction are sufficient to determine the speed of light.

Ampere discovered that two long parallel wires carrying
electric currents in the same direction attract each other magnetically, the
force per unit length being proportional to the product of the currents (so
oppositely directed currents repel) and decaying with distance as 1/*r*.
In modern (SI) notation, his discovery is written (*F* in Newtons)

The modern convention is that the constant appearing here is *exactly *10^{-7},
this *defines *our *present* unit of current, the *ampere. *To repeat:
is not something to measure experimentally, it's just a funny way of
writing the number 10^{-7}! That's not quite fairit
has dimensions to ensure that both sides of the above equation have the same
dimensionality. (Of course, there's a historical reason for this strange
convention, as we shall see later). Anyway,
if we bear in mind that dimensions have been taken care of, and *just* write
the equation

it's clear that this defines the unit currentone
**ampere**as
that current in a long straight wire which exerts a magnetic force of newtons per meter of wire on a parallel wire
one meter away carrying the same current.

However, after we have established our unit of currentthe
ampere*we
have also thereby defined our unit of charge*, since current is a flow of
charge, and the unit of charge must be the amount carried past a fixed point in
unit time by unit current. Therefore,
our unit of chargethe
**coulomb***is
defined by stating that a **one amp *current in a wire carries *one
coulomb per second *past a fixed point.

To be consistent, we must do electrostatics using this
*same *unit of charge. Now, the electrostatic force between two charges
is The constant appearing here, now written ,
*must be experimentally measured*its
value turns out to be .

To summarize: to
find the value of ,
*two* experiments have to be performed.
We must first establish the unit of charge from the unit of current by
measuring the magnetic force between two current-carrying parallel wires. Second, we must find the electrostatic force
between measured charges. (We could, alternatively, have defined some other
unit of current from the start, then we would have had to find both and by experiments on magnetic and electrostatic
attraction. In fact, the ampere *was *originally
defined as the current that deposited a definite weight of silver per hour in
an electrolytic cell).

We established earlier in the course that the total flux of electric field out of a closed surface is just the total enclosed charge multiplied by ,

This is **Maxwell’s
first equation**. It
represents completely covering the surface with a large number of tiny patches
having areas .
(The little areas are small enough to be
regarded as flat, the vector magnitude *dA*
is just the value of the area, the direction of the vector is perpendicular to
the area element, pointing outwards away from the enclosed volume.) Hence the dot product with the electric field
selects the component of that field pointing perpendicularly outwards (it would
count negatively if the field were pointing inwards)this
is the only component of the field that contributes to actual electric flux across
the surface. (Remember flux just means
flowthe
picture of the electric field in this context is like a fluid flowing out from
the charges, the field vector representing the direction and velocity of the
flowing fluid.)

The second Maxwell equation is the analogous one for
the magnetic field, which has no sources or sinks (no magnetic monopoles, the
field lines just flow around in closed curves). Again thinking of the force lines as
representing a kind of fluid flow, the so-called "magnetic flux", we
see that for a closed surface, as much magnetic flux flows into the surface as
flows outsince
there are no sources. This can perhaps
be visualized most clearly by taking a group of neighboring lines of force
forming a slender tubethe
"fluid" inside this tube flows round and round, so as the tube goes
into the closed surface then comes out again (maybe more than once) it is easy
to see that what flows into the closed surface at one place flows out at
another. Therefore the net flux out of the enclosed volume is zero, **Maxwell’s second equation**:

The first two Maxwell's equations, given above, are for
integrals of the electric and magnetic fields over *closed surfaces*. Maxwell's other two equations, discussed
below, are for integrals of electric and magnetic fields around *closed
curves *(taking the component of the field pointing along the curve). These
represent the work that would be needed to take a charge around a closed curve
in an electric field, and a magnetic monopole (if one existed!) around a closed
curve in a magnetic field.

The simplest version of **Maxwell's third equation**
is for the special **electrostatic**
case:

The path integral for electrostatics*.*

However, we know that this is only part of the truth,
because from Faraday's Law of Induction, if a closed circuit has a *changing
magnetic flux *through it, a circulating current will arise, which means
there is a *nonzero voltage *around the circuit.

The **complete Maxwell's third equation**

where the area integrated over on the right hand side spans the path (or circuit) on the left hand side, like a soap film on a loop of wire. (The best way to figure out the sign is to use Lenz’ law: the induced current will generate a magnetic field opposing the changing of the external field, so if an external upward field is decreasing, the current thereby generated around the loop will give an upward pointing field.)

It may seem that the integral on the right hand side
is not very clearly defined, because if the path or circuit lies in a plane,
the natural choice of spanning surface (the "soap film") is flat, but
how do you decide what surface to choose to do the integral over for a wire
bent into a circuit that doesn’t lie in a plane? The answer is *that it doesn’t matter what surface you choose*, as long as the wire
forms its boundary. Consider two
different surfaces both having the wire as a boundary (just as both the
northern hemisphere of the earth’s surface and the southern hemisphere have the
equator as a boundary). If you add these
two surfaces together, they form a single closed surface, and we know that for
a closed surface .
This implies that for one of the two surfaces bounded by the
path is equal to for the other one, so that the two will add to
zero for the whole closed surface. But
don’t forget these integrals for the whole closed surface are defined with the
little area vectors pointing outwards from the enclosed volume. By imagining
two surfaces spanning the wire that are actually close to each other, it is
clear that the integral over one of them is equal to the integral over the
other if we take the vectors to point in the same direction for
both of them, which in terms of the enclosed volume would be outwards for one
surface, inwards for the other one. The bottom line of all this is that the
surface integral is the same for any surface spanning the path,
so it doesn’t matter which we choose.

The equation analogous to the electrostatic version of the
third equation given above, but for the magnetic field, is **Ampere's law***,*

*for the magnetostatic case*,

where the currents counted are those threading through the path we're integrating around, so if there is a soap film spanning the path, these are the currents that punch through the film (of course, we have to agree on a direction, and subtract currents flowing in the opposite direction).

We must now consider whether this equation, like the
electrostatic one, has limited validity. In fact, it was not questioned for a
generation after Ampere wrote it down: Maxwell's great contribution, in the
1860's, was to realize that it was *not *always valid.

A simple example to see that *something *must
be wrong with Ampere's Law in the general case is given by Feynman in his *Lectures
in Physics **(II,
18-3)*. Suppose we use a
hypodermic needle to insert a spherically symmetric blob of charge in the
middle of a large vat of solidified jello (which we assume conducts
electricity). Because of electrostatic
repulsion, the charge will dissipate, currents will flow outwards in a
spherically symmetric way. *Question*: does this outward-flowing
current distribution generate a magnetic field? The answer must be *no *, because since
we have a completely spherically symmetric situation, it could only generate a
spherically symmetric magnetic field. But
the only possible such fields are one pointing outwards everywhere and one
pointing inwards everywhere, both corresponding to non-existent monopoles. So, there can be no magnetic field.

However, imagine we now consider checking Ampere's law by taking as a path a horizontal circle with its center above the point where we injected the charge (think of a halo above someone’s head.) Obviously, the left hand side of Ampere's equation is zero, since there can be no magnetic field. On the other hand, the right hand side is most definitely not zero, since some of the outward flowing current is going to go through our circle. So the equation must be wrong.

Ampere's law was established as the result of large
numbers of careful experiments on all kinds of current distributions. So how could it be that something of the kind
we describe above was overlooked? The
reason is really similar to why electromagnetic induction was missed for so
long. No-one thought about looking at *changing
*fields, all the experiments were done on steady situations. With our ball of charge spreading outward in
the jello, there is obviously a changing electric field. Imagine yourself in the jello near where the
charge was injected: at first, you would feel a strong field from the nearby
concentrated charge, but as the charge spreads out spherically, some of it
going past you, the field will decrease with time.

Maxwell himself gave a more practical example: consider Ampere's law for the usual infinitely
long wire carrying a steady current *I *, but now break the wire at some
point and put in two large circular metal plates, a capacitor, maintaining the
steady current *I *in the wire everywhere else, so that charge is simply
piling up on one of the plates and draining off the other.

Looking now at the wire some distance away from the
plates, the situation appears normal, and if we put the usual circular path
around the wire, application of Ampere's law tells us that the magnetic field
at distance *r *, from

is just

* *

(Reminder on field direction: the right hand ruleif you curl the fingers of your right hand around an imaginary wire, a current flowing in the direction indicated by your thumb will generate circular magnetic field lines in the direction indicated by your fingers.)

Recall, however, that we defined the current threading the path in terms of current punching through a soap film spanning the path, and said this was independent of whether the soap film was flat, bulging out on one side, or whatever. With a single infinite wire, there was no escapeno contortions of this covering surface could wriggle free of the wire going through it (actually, if you distort the surface enough, the wire could penetrate it several times, but you have to count the net flow across the surface, and the new penetrations would come in pairs with the current crossing the surface in opposite directions, so they would cancel).

Once we bring in Maxwell's parallel plate capacitor,
however, there *is *a way to distort the surface so that no current
penetrates it at all: we can run it between the plates!

The question then arises: can we rescue Ampere's law by
adding another term just as the electrostatic version of the third equation was
rescued by adding Faraday's induction term? The answer is of course yes: although there is no *current *crossing
the surface if we put it between the capacitor plates, there is certainly a *changing
electric field *, because the capacitor is charging up as the current *I
*flows in. Assuming the plates are close together, we can take all the
electric field lines from the charge *q*
on one plate to flow across to the other plate, so the total electric flux
across the surface between the plates,

Now, the current in the wire, *I *, is just the rate
of change of charge on the plate,

* *

Putting the above two equations together, we see that

Ampere's law can now be written in a way that is correct no matter where we put the surface spanning the path we integrate the magnetic field around:

This is **Maxwell’s
fourth equation**.

Notice that in the case of the wire, *either *the
current in the wire, *or *the increasing electric field, contribute on
the right hand side, depending on whether we have the surface simply cutting
through the wire, or positioned between the plates. (Actually, more complicated
situations are possiblewe
could imaging the surface partly between the plates, then cutting through the
plates to get out! In this case, we
would have to figure out the current actually in the plate to get the right
hand side, but the equation would still apply).

Maxwell referred to the second term on the right hand
side, the changing electric field term, as the "displacement
current". This was an analogy with
a dielectric material. If a dielectric
material is placed in an electric field, the molecules are distorted, their
positive charges moving slightly to the right, say, the negative charges
slightly to the left. Now consider what
happens to a dielectric in an *increasing *electric field. The positive charges will be displaced to the
right by a continuously increasing distance, so, as long as the electric field
is increasing in strength, these charges are moving: there is actually a *displacement
current *. (Meanwhile, the negative
charges are moving the other way, but that is a current in the same direction,
so adds to the effect of the positive charges' motion.) Maxwell's picture of the vacuum, the aether,
was that it too had dielectric properties somehow, so he pictured a similar
motion of charge in the vacuum to that we have just described in the
dielectric. This is why the changing
electric field term is often called the "displacement current", and
in Ampere's law (generalized) is just added to the real current, to give
Maxwell's fourthand
finalequation.

Going back for a moment to Ampere's law, we stated it as:

for magnetostatics

where the currents counted are those threading through the path we’re integrating around, so if there is a soap film spanning the path, these are the currents that punch through the film. Our mental picture here is usually of a few thin wires, maybe twisted in various ways, carrying currents. More generally, thinking of electrolytes, or even of fat wires, we should be envisioning a current density varying from point to point in space. In other words, we have a flux of current and the natural expression for the current threading our path is (analogous to the magnetic flux in the third equation) to write a surface integral of the current density over a surface spanning the path, giving for magnetostatics

path integral , (surface integral, over surface spanning path)

The question then arises as to whether the surface integral
we have written on the right hand side above depends on *which *surface
we choose spanning the path. From an argument exactly parallel to that for the
magnetic flux in the third equation (see above), this will be true *if and
only if ** *for a *closed surface *(with the
path lying in the surfacethis
closed surface is made up by combining two different surfaces spanning the
path).

Now, taken over a *closed *surface is just
the net current flow out of the enclosed volume. Obviously, in a situation with
steady currents flowing along wires or through conductors, with no charge
piling up or draining away from anywhere, this is zero. However, if the total
electric charge *q *, say, enclosed by the closed surface *is *changing
as time goes on, then evidently

where we put in a minus sign because, with our convention, is a little vector pointing *outwards*,
so the integral represents net flow of charge *out *from the surface,
equal to the rate of *decrease *of the enclosed total charge.

To summarize: if the local charge densities are
changing in time, that is, if charge is piling up in or leaving some region,
then over a closed surface around that region. That
implies that over one surface spanning the wire will be *different
*from over another surface spanning the wire if
these two surfaces together make up a closed surface enclosing a region
containing a changing amount of charge.

The key to fixing this up is to realize that although it can be written as another surface integral
over the same surface, using the *first *Maxwell equation, that
is, the integral over a closed surface

where *q* is the
total charge in the volume enclosed by the surface.

By taking the time rate of change of both sides, we find

* *

Putting this together with gives:

* *

for *any closed *surface, and consequently this is a
surface integral that must be the same for *any *surface spanning the
path or circuit! (Because two different
surfaces spanning the same circuit add up to a closed surface. We’ll ignore the
technically trickier case where the two surfaces intersect each other, creating
multiple volumesthere
one must treat each created volume separately to get the signs right.)

Therefore, this is the way to generalize Ampere's law from the magnetostatic situation to the case where charge densities are varying with time, that is to say the path integral

and this gives the same result for any surface spanning the path.

As a preliminary to looking at electromagnetic waves, we consider the magnetic field configuration from a sheet of uniform current of large extent. Think of the sheet as perpendicular to this sheet of paper, the current running vertically down into the paper. It might be helpful to visualize the sheet as many equal parallel fine wires uniformly spaced close together, carrying equal (small) currents:

...................................................................................... (wires)

The magnetic field from this current sheet can be found using Ampere's law applied to a rectangular contour in the plane of the paper, with the current sheet itself bisecting the rectangle, so the rectangle's top and bottom are equidistant from the current sheet in opposite directions.

Applying Ampere’s law to the above rectangular
contour, there are contributions to (taken clockwise) only from the top and
bottom, and they add to give *2BL* if the rectangle has side *L*. The
total current enclosed by the rectangle is *IL*, taking the current
density of the sheet to be *I* amperes per meter (how many little wires
per meter multiplied by the current in each wire).

Thus, immediately gives:

*B = µ _{0}I/2*

a magnetic field strength *independent *of distance *d
*from the sheet. (This is the magnetostatic analog of the electrostatic
result that the electric field from an infinite sheet of charge is independent
of distance from the sheet.) In real life, where there are no infinite sheets
of anything, these results are good approximations for distances from the sheet
small compared with the extent of the sheet.

Consider now how the magnetic field develops if the current
in the sheet is suddenly switched on at time *t *= 0. We will assume that
sufficiently close to the sheet, the magnetic field pattern found above using
Ampere's law is rather rapidly established.

In fact, we will assume further that the magnetic field
spreads out from the sheet like a tidal wave, moving in both directions at some
speed *v *, so that after time *t *the field within distance *vt *of
the sheet is the same as that found above for the magnetostatic case, but
beyond *vt *there is at that instant no magnetic field present.

Let us now apply Maxwell's equations to this guess to see if
it can make sense. Certainly Ampere's law doesn't work by itself, because if we
take a rectangular path as we did in the previous section, for *d *< *vt
*everything works as before, but for a rectangle extending *beyond* the spreading magnetic field, *d
*> *vt *, there will be *no*
magnetic field contribution from the top and bottom of the rectangle, and
hence

but there is definitely enclosed current!

We are forced to conclude that for Maxwell's fourth equation
to be correct, *there must also be a changing electric field through the
rectangular contour.*

Let us now try to nail down what this electric field through
the contour must look like. First, it
must be through the contour, that is, have a component perpendicular to the
plane of the contour, in other words, perpendicular to the magnetic field. In fact, electric field components in other
directions won't affect the fourth equation we are trying to satisfy, so we
shall ignore them. Notice first that for
a rectangular contour with *d *< *vt*, Ampere's law works, so we *don't*** **want
a changing electric field through such a contour (but a constant electric field
would be ok).

Now apply Maxwell's fourth equation to a rectangular contour
with *d *> *vt,*

It is: path integral (over surface spanning path).

For the rectangle shown above, the integral on the left hand side is zero because is perpendicular to along the sides, so the dot product is zero, and is zero at the top and bottom, because the outward moving "wave" of magnetic field hasn’t gotten there yet. Therefore, the right hand side of the equation must also be zero.

We know , so we must have:

So, as long as the outward moving front of magnetic field,
travelling at *v *, hasn't reached the top and bottom of the rectangular
contour, the electric field through the contour *increases linearly with time*, but the increase drops to zero
(because Ampere's law is satisfied) the moment the front reaches the top and
bottom of the rectangle. The simplest way to get this behavior is to have an
electric field of strength E, perpendicular to the magnetic field, everywhere
there is a magnetic field, *so the electric field also spreads outwards at
speed v**.* (Note that, unlike the magnetic field, the
electric field must point the same way on both sides of the current sheet,
otherwise its net flux through the rectangle would be zero.)

After time *t *, then, the electric field flux through
the rectangular contour (in the *yz*-plane
in the diagram above) will be just field x area = *E.*2*.vtL *, and
the rate of change will be 2*EvL *. (It's spreading both ways, hence the
2).

Therefore *ε _{0}*

Since *B *= *µ _{0}I*/2, this
implies:

*B* = *µ _{0}ε_{0}vE.*

But we have another equation linking the field strengths of the electric and magnetic fields, Maxwell's third equation:

We can apply this equation to a rectangular contour with
sides parallel to the *E *field, one
side being within *vt* of the current
sheet, the other more distant, so the only contribution to the integral is *EL* from the first side, which we take to
have length *L*. (This contour is all on one side of the
current sheet.) The area of the rectangle the magnetic flux is passing through
will be increasing at a rate *Lv*
(square meters per second) as the magnetic field spreads outwards.

It follows that

*E = vB*.

Putting this together with the result of the fourth equation,

*B = µ _{0}ε_{0}vE*,

we deduce

*v ^{2}* = 1/

Substituting the defined value of *µ _{0}*, and the experimentally measured value of

To understand how this relates to wave propagation, imagine now that shortly after the current is switched on, the value of the current is suddenly doubled. Repeating the argument above for this more complicated situation, we find the following scenario:

We could have ramped up the field in a series of steps —and the profile of the magnetic and electric fields would, effectively, be a graph of how the current built up over time.

The next step is to imagine an electric current in the sheet that’s oscillating like a sine wave as a function of time: the magnetic and electric fields will evidently be sine waves too! In fact, this is how electromagnetic waves are generated. Of course, there’s no such thing as an infinite current sheet, an antenna has an oscillating current going up and down a wire. But the mechanism is essentially the same: the only difference is that the geometry of the waves is complicated. Far away, they’ll look like expanding spheres, a three-dimensional version of the ripples on a pond when a stone falls in, instead of propagating planes. But at large distances a small fraction of these expanding spheres w, and that looks like a series of planes. The picture above of how the electric and magnetic fields relate to each other and to the direction of propagation of the wave is correct.

This is how Maxwell discovered a speed equal to the speed of light from a purely theoretical argument based on experimental determinations of forces between currents in wires and forces between electrostatic charges. This of course led to the realization that light is an electromagnetic wave, and that there must be other such waves with different wavelengths. Hertz detected other waves, of much longer wavelengths, experimentally, and this led directly to radio, tv, radar, cellphones, etc.