The first great synthesis of ideas in physics took place in 1666 when Issac Newton
realised that the force which causes apples to fall downwards is the same as the
force which maintains the planets in elliptical orbits around the Sun. The second
great synthesis, which we are about to study in more detail, took place in
1830 when Michael Faraday discovered that electricity and magnetism are two
aspects of the same thing, usually referred to as
*electromagnetism*. The third great synthesis, which we shall discuss
presently, took place in 1873 when James Clerk Maxwell demonstrated that light
and electromagnetism are intimately related. The last (but, hopefully,
not the final) great synthesis took place in 1967 when Steve Weinberg and
Abdus Salam showed that the electromagnetic force
and the weak nuclear force (*i.e.*, the one which is responsible for decays)
can be combined to give the electroweak force.
Unfortunately, Weinberg's work lies well beyond the scope of this lecture course.

Let us now consider Faraday's experiments, having put them in their proper
historical context.
Prior to 1830, the only known way to make an electric
current flow through a conducting wire was to connect the ends of the wire to
the positive and negative
terminals of a battery. We measure a battery's ability to push current
down a wire in terms of its *voltage*, by which we mean the voltage difference
between its positive and negative terminals. What does voltage correspond
to in physics?
Well, volts are the units used to measure electric scalar potential, so when we
talk about a 6V battery, what we are really saying is that the difference in
electric scalar potential between its positive and negative terminals is six volts.
This insight allows us to write

(370) |

Let us now consider a closed loop of wire (with no battery). The voltage around such a loop, which is sometimes called the *electromotive
force* or *e.m.f.*, is

(371) |

(372) |

Faraday continued his experiments and found that
another way of generating an electromotive force around a loop of wire
is to keep the magnetic field constant
and move the loop. Eventually, Faraday was able to
formulate a law which accounted for all of his experiments. The e.m.f. generated around a loop of wire in a magnetic field is proportional to
the rate of change of the flux of the magnetic field through the loop. So,
if the loop is denoted , and is some surface attached to the loop, then Faraday's
experiments can be summed up by writing

S.I. units have been carefully chosen so as to make in
the above equation. The only thing we now have to decide is whether
or . In other words, which way around the loop does the induced e.m.f. want to drive the current? We possess a general principle which allows us to
decide questions like this. It is called
*LeChatelier's principle*. According to Le Chatelier's principle, every change
generates a reaction which tries to minimize the change. Essentially, this means
that the Universe is stable to small perturbations. When this principle
is applied to the special case of
magnetic induction, it is usually called *Lenz's law*. According to Lenz's
law, the current induced around a closed loop
is always such that the magnetic field it produces tries to counteract the
change in magnetic flux which generates the electromotive force.
From Fig. 34, it is clear that if the magnetic field is
increasing and the current circulates clockwise (as seen from above) then
it generates a field which opposes the increase in magnetic flux
through the loop, in
accordance with Lenz's law. The direction of the current is opposite to the
sense of the current loop (assuming that the flux of through the
loop is positive), so this implies that in Eq. (373). Thus, Faraday's
law takes the form

Experimentally, Faraday's law is found to correctly predict the e.m.f. (*i.e.*,
) generated in any wire loop, irrespective of
the position or shape of the loop.
It is reasonable to assume that the same e.m.f. would be
generated in the absence of the wire (of course, no current would flow
in this case). Thus, Eq. (374) is valid for any closed loop . If Faraday's
law is to make any sense then it must also be true for any surface attached to the
loop . Clearly, if the flux of the magnetic field through the loop depends on
the surface upon which it is evaluated then Faraday's law is going to predict
different e.m.f.s for different surfaces. Since there is no preferred surface for
a general non-coplanar loop, this would not make very much sense. The condition
for the flux of the magnetic field,
, to depend
only on the loop to which the surface is attached, and not on the nature
of the surface itself, is

Faraday's law, Eq. (374), can be converted into a field equation using
Stokes' theorem. We obtain

(377) |

The divergence of Eq. (376) yields

(378) |