So to understand electromagnetic
induction quantitatively, we have to
understand magnetic flux. And flux is not
a particularly alien concept; you're
worrying about flux whenever you're
trying to put your solar panels on your
roof so they catch the most sunlight. And
you're worrying about flux whenever
you're trying to avoid getting wet in
the rain. In general flux is the flow of
some physical quantity. So one example might be rain. Rain comes
down comes down in a particular
direction and it comes down sometimes in
different directions in different places.
And also different amounts in different
places. So you might have a bit where
it's raining heavily and hard and going
in that direction, and then a bit where
it's raining quite lightly. And more
straight down. And so rain is described
as a vector field, which remember it's
just a vector at each point in space. And
what are the units of this vector? Well
it's a certain amount of water which I
guess we'd measure in kilograms, your SI unit,
per second per square metre. So if we
have a football field, we'll gather more
water than if we have a small glass. And
so this vector field has units of
kilograms of water per second per square
metre. So if we wait for an hour we'll
get more water than if we wait for one
second. So the other example we talked
about with sunlight. Now sunlight comes
in pretty uniformly at a particular
angle. If we're out in space looking at a
really big scale of course it would be
spreading out in all directions. But down
here on earth it comes in pretty evenly.
So that's also a vector field and what
are the units of that? Well, sunlight gives
us energy, so we have a certain amount of
energy and again we have a certain amount
of energy per second and per square
metre. Which also the same thing as a
watt per square metre. So you might be
trying to catch all this rain into a
rain tank so that you can avoid the
horrible drought coming up. Or you might
be trying to catch this sunlight on a
solar panel so you can free yourself
from energy costs. Or you might be trying
to find out how much magnetic field is
going through a loop of wire, or possibly
even multiple loops of wire. The total
flux going through each of these three
areas is going to depend on a few things.
It's going to depend on the strength of
the vector field, so if you have more
watts per square metre, you're going to
get more energy coming through. If you
have more kilograms per second per
square metre, you can have more water
coming through. And if you have more
magnetic field, you can have more
magnetic flux coming through. Secondly
it's going to depend on the size of the
area: the bigger the area, the more flux
you're going to have coming through. And
finally it's also going to depend on the
angle. So doesn't matter if you have an
absolutely enormous solar panel, if you
angle it wrong and the sunlight misses
it goes straight past, it just hits the edge. So it's
not exactly the size of the area that
matters, it's the size of the area
perpendicular to the vector field. So
it's really this area here (insert ref) that matters,
not the area of the actual solar cell.
So the magnetic flux through the coil is
usually defined by the symbol capital
Phi, and it's just equal to the magnetic
field times the perpendicular area.
Now if the magnetic field changes
direction through your coil, then you've
got a more complicated thing to do
because you have to break your area up
into little pieces of area such that you
had a piece of area so small the
magnetic field is basically going in
constant direction. And then you can
calculate the flux to that little piece
of area and then add up all those pieces
of flux. But we won't do things are that
complicated in calculation. So coming
back to electromagnetism, remember that
what Faraday found was the current was
generated by a changing magnetic field.
And after more detailed experiments and
analysis it was found that the
electromagnetic force, in other words the
voltage around that loop, was equal to
the rate of change of the magnetic flux.
Until use the capital Delta describe a
change as before, and we used the B to
denote the magnetic flux and then we've got this Delta Phi over Delta T; that's the
rate of change of magnetic flux. And so
that voltage will be applied to every
loop that we have in this coil. And so if
we have multiple windings of that coil,
we're going to get that voltage for each
one of them. And so the total voltage is
going to be just the number of loops
times the rate of change of magnetic
flux.
And that is now known as Faraday's law
