in this video we get quantitative with the induced emf that occurs when the flux through
a loop changes we start by quantifying magnetic flux then we state faraday's law for the size of
the induced emf when the flux changes over a known period of time so in the picture we have magnetic
field lines penetrating a loop and flux is really a way of talking about how many field lines we've
captured in the loop note that it matters which direction they're going through the loop so we use
a normal vector that's this green vector labeled as n hats we use a normal vector to orient the
surface and in this case field lines are passing through the loop almost in the same direction as
the normal vector but there is an angle of theta between them now if theta was zero then the field
would be perpendicular to the face of the loop and we will maximize our flux if theta is 90 degrees
that means the face is actually parallel to the magnetic field lines and we'll get zero flux
so we set out to quantify the magnetic flux and we think about the following proportionalities
first we expect that magnetic flux should be proportional to the field
strength so if i doubled the intensity of the magnetic field it would double the flux
going to use the symbol capital phi for flux and i'm saying phi is proportional to b
second flux should be proportional to area if i doubled the area captured in a loop and kept
the magnetic field the same then of course we'd capture twice the magnetic field lines and third
flux should be proportional to the cosine of that angle theta when that angle theta is zero
the normal vector points in the same direction as the magnetic field in other words the surface is
perpendicular to the magnetic field lines and will capture the maximum amount of field lines
if we put these three proportionalities together we get the correct definition for magnetic flux
next we introduce faraday's law quantifying the induced emf when the flux changes recall
from lenses law that the basic idea here is that the loop will oppose any change in flux
by inducing or attempting to induce a current that creates
field lines through the loop in a direction opposing the change for example if the initial
magnetic field was zero through the loop and you suddenly turn the magnetic field on
the loop will generate magnetic field lines in the opposite direction to try to keep it at zero
conversely if you originally have a magnetic field through the loop and you turn it off
a current will be induced in the loop to try to sustain that original direction of flux
you can summarize this by saying nature abhors a change in flux in other words the loop's going to
try to do whatever it can to keep the flux through it exactly the same and again that will be an
induced current if the circuit is complete if you have a loop where it's disconnected and current
can't flow you'll still see an induced emf that's trying to make current flow in that direction
so exactly how large is this induced emf that's faraday's law and to quickly motivate faraday's
law we're going to look at a few proportionalities just like we did for the definition of flux first
the induced emf should be proportional to the change in flux if i have a bigger change in
flux then of course it's going to create a bigger emf to try to counteract that change
second the induced emf is going to be inversely proportional to the time it takes
to change that flux so if you were to change the flux very very slowly
it would take a very very small current in order to oppose the change in flux
a very fast change in flux is going to require a large emf and a large current to counteract the
change so i hope that makes sense intuitively and third the induced emf is going to be proportional
to how many turns there are in the loop so now we're generalizing to the possibility that our
loop has many many turns in it and not just one single coil of wire so if i double the number of
turns in a coil that doubles the flux just because i have twice as many loops catching field lines
and so of course if i change the flux i'm going to get twice the emf induced
and so we arrive at faraday's law and there's a minus sign in this just indicating that lenses law
direction that the induced emf will be acting in a direction that fights the change in flux okay
so now we know how to calculate magnetic flux and we know how to calculate the induced emf
if we change the flux through a loop let's wrap things up with a simple example so in this example
i have a coil of wire that has actually 200 turns in it and i have the wire disconnected at its ends
so these are referred to in the problem as the terminals of the coil and we're suddenly switching
on a one tesla magnetic field through this thing the process of turning on that field takes 0.1
seconds it's totally perpendicular to the face of the coil so i don't have to worry about the cosine
of theta here theta is going to be zero so my flux is maximized and my change in flux is maximized
with respect to the angle and i have 200 turns in this coil i'm given a coil diameter of 20
centimeters i'm asked first to get the average induced emf across the terminals
so again current is not actually going to be able to flow here but i'll still see an induced voltage
across the terminals that's attempting to get the current flowing in the direction that opposes the
change in flux and then i said in the second part if the terminals are bridged with a wire
and the coil has a total resistance of 8 ohms give me the average current that
would flow through the coil all right so let's start by actually computing
the new flux through the coil after the field has been turned on again that's a b
cosine of theta where theta is the angle between the normal vector and the magnetic field
the angle is zero so the cosine is 1. so i need the area of the loop that's pi r squared or
you could call it one quarter pi d squared either way i'm going to go ahead and find
radius in this case that's 10 centimeters and the magnetic field was just one tesla
so i get .0314 tesla meters squared for the final flux and of course the initial was zero because
the magnetic field was off initially then we apply faraday's law and we say e induced
we don't really care about the minus sign here but i'll put it in at first
it's going to be negative n delta phi delta t again the minus sign just indicates the
direction of this the induced emf is going to try to produce a current that opposes
the change in flux through the loop and we have 200 turns so capital n is 200.
my change in flux it started at zero and ended at .0314 so my change in flux is .0314
and the time it took is one tenth of a second so i get negative 62.8 volts for this emf
again the minus sign just indicating the direction it's just as correct in this case
to just say okay my induced emf is 62.8 volts and we can agree the direction is given by lenses law
all right so next i tell you that the total resistance of the wire in this coil is 8 ohms
and i know that over this 0.1 second period i have this average induced voltage of 62.8 volts
so i should be able to get the current induced in the loop
that's given by ohm's law i equals v over r 62.8 volts over 8 ohms and this gives me 7.85 amps