in this video we're looking at a coil of wire rotated in a constant magnetic field so i've
created this magnetic field by bringing the north pole of a magnet up to the left-hand side
south pole on the right-hand side creating a uniform rightward magnetic field
and the loop is turning around on an axis like this in this magnetic field so the magnetic
field through the coil is constantly changing and that means we're going to generate an emf
that's constantly changing to oppose the changes in magnetic field through the coil
and this generated emf actually turns out to be sinusoidal so this is the basic operating
principle of an electric generator it's just coils of wire rotating in a strong magnetic field
so we're going to draw the directions of induced current at each point in the rotation just to
get a qualitative sense for what's going on and then we're going to derive a formula for the emf
as a function of time and finally we work a simple example of how to find the peak and rms ac voltage
produced by a rotating coil in a magnetic field so in the picture we're going to say that this coil
is rotating with an angular velocity of omega and remember omega is measured in radians per second
i'll just put that in at the bottom as a reminder the units of omega are radians per second
well if i took omega and multiplied by time that would give me an angle with respect to
the starting value so that's what i've done in the overhead view in the picture on the right radians
per second times seconds gives me radians and i'm going to assume my starting point here is theta
equals zero when the normal vector perpendicular to the surface of the coil was actually perfectly
aligned with the magnetic field after that my angle at any moment in time is given by omega t
so we can use lenses law to find the direction of the induced current in the coil at any point
in time in this picture so we're going to talk through these step by step and get all the induced
currents written in here so i have some initial orientation where the normal vector is pointing in
exactly the same direction as the magnetic field so that means magnetic flux is coming rightward
through the loop in the same direction as the normal vector and this loop is used to having flux
coming through the same direction as the normal vector so what happens in the second picture
when i've rotated my loop so that the normal vector points into the page that loop is now
missing all of those field lines that are coming through the same direction as the normal vector
and current is going to flow in a way that makes up for the change in flux
trying to maintain the original field direction so my induced magnetic field will be in the
same direction as the normal vector then i use the ampere's law right hand rule to figure out
which way does the current have to flow in order to induce that inward magnetic field
so i want to be able to point my thumb in the direction of current and have my fingers in my
right hand wrap around the same direction as that induced magnetic field and in this
case that current must be looping around in the clockwise direction in order to get this done
so that's my induced current after that first 90 degree flip and keep in mind this is only a
transient phenomenon and it's only a partial cancellation of the change in flux and i'm
going to say very quickly my loop is used to having no flux lines pointing through it
but then i quickly flip it another 90 degrees and it started out with very little flux through
it and now it has a bunch of flux opposing the direction of the normal vector well this means
current has to be induced to try to get the flux through the loop back to zero using the
right hand rule again i get induced current in the same direction as it did in the previous picture
so where i labeled i in the previous picture this is now the arrow on the near side of the
coil facing us in the third picture put my thumb in the direction of that current
loop the fingers of my right hand around it creates flux out of the loop to try to oppose
that change of new magnetic field lines going through the loop opposite to the normal vector
so now let's say the loop is used to having all these field lines opposing
the direction of the normal vector and i rotate into the position of the last picture
the normal vector is now pointing out at us and that means there's no magnetic flux
through the loop from the permanent magnet but my coil was used to having flux coming
in opposing the direction of the normal vector so i need to generate current in a direction
that creates magnetic field in the opposite direction of the normal vector that is now
going to be the opposite direction as what it was in the previous picture again i'm just following
the same spot on the loop to try to get the 3d visualization to work i had this little section
right here with a current moving clockwise here it was moving up when the loop was rotated towards us
and then here suddenly that current is moving down so i have a reversal in the direction of
the current i put my thumb in the direction of the current wrap my fingers around using
the ampere's law right hand rule and the induced magnetic field is into the loop in other words
opposing the direction of the normal vector to try to keep things the way they were in the prior step
with magnetic field lines opposing the normal vector direction just to wrap things up the
loop the loop has almost completed one full rotation and we'll be back to the original picture
so i flip it another 90 degrees and suddenly i have a bunch of new flux pointing in the same
direction as the normal vector the loop is going to try to oppose that and it creates flux opposing
the direction of the normal vector and again following that same little segment on the wire
that's an induced downward current on this section of the wire that is far from us in this picture
so this is the far side not the near side i put my thumb in the direction of that current wrap
the fingers of my right hand around and that's an induced flux opposing the direction of the normal
vector attempting to cancel out all that new flux in the same direction as the normal vector
so there's qualitatively what happens i end up with a current flipping back and forth in the loop
as it's fighting all these changes in flux from the exterior magnetic field
that we're imposing on it so now we're going to get quantitative with the induced emf
first we want to write the flux through one turn of the loop as a function of theta
and that's given by the flux formula the flux is a b times cosine theta where theta is the angle
between the normal vector and the direction of the magnetic field then using faraday's
law we're trying to compute the rate of change in flux to get the induced emf so
here's how i've written faraday's law previously induced emf is negative
n times the rate of change in the flux now this is one case where using finite changes
is not going to be very effective and i'm going to rewrite this as a derivative so we're going
to do just a little bit of calculus here i'm going to write this as negative n d
phi dt and again my goal here is to find the induced emf as a function of time so what i want
is a time derivative of the flux well remember the flux is a function of time because my angle is a
function of time and that's given by omega t so the flux could be written as a b cosine omega t
well that means my induced voltage is given by negative n times the time derivative
of a b cosine omega t and the a and b there are spectators they're constants
they can move right outside the derivative and then i need the time derivative of cosine omega t
well that's negative sine omega t but then the chain rule forces me to tack on the derivative of
omega t with respect to t which is omega finally i end up with my induced emf as a function of time n
a b omega sine omega t and i see that for a loop rotating at a constant rate in a magnetic field
i actually do get a sinusoidal induced voltage so this is the
natural output of an electric generator a sinusoidal voltage where that coefficient
of sine omega t i could call that epsilon zero and that would be the peak voltage
there are a couple important qualitative points about this formula first if n is bigger then my
induced voltage is bigger hopefully that makes sense if you have more turns in your coil then
you're dealing with bigger induced voltages just because you have more loops to experience
flux changes if i have a bigger area i get bigger flux changes and therefore bigger induced voltage
if i have a bigger magnetic field of course that increases the flux and the flux changes so i get
a bigger induced voltage and if i have a bigger angular velocity in other words the spin rate in
radians per second well that means the fluxes are changing faster i'm making those delta t's smaller
so i get a bigger induced voltage so that peak voltage is proportional to all of these things
so there's just one last point i want to make on this slide and then we'll wrap
things up with an example the problems that we solve involving this generator formula
will often give you the frequency of the rotation or the period of the rotation and you've got to be
able to relate that to omega so i remember for a sinusoidal function that the period
is 2 pi divided by the coefficient of t so there's the relationship to omega
sometimes it's useful to solve for omega and that say omega is 2 pi divided by the period
just depends on how the problem is given to you and then frequency is
the reciprocal of periods so that must be omega over 2 pi
and sometimes it's useful again to solve for omega and write it as 2 pi times the frequency
so when we're working these generator formula problems you're frequently dealing with these
sorts of conversions i'll just leave them here as a side note and then we'll wrap things up
with a little example so in our example we have a circular coil of diameter 10 centimeters diameter
10 centimeters not radius and it has 100 turns and it rotates at 500 rotations per minute or rpms
in a constant magnetic field of half a tesla so immediately we're dealing with this rotation rate
conversion issue i have 500 rotations per minute and i want to get this in radians per second
so i wasn't given a frequency or period this time i was just given rotations per minute
so it's a little different conversion and i get 2 pi radians for rotation just doing
some unit analysis here and then convert those minutes so i have one minute for every
60 seconds my only surviving units here are radians
in the numerator seconds in the denominator so that's good and i get 52.4 radians per second
again that's omega from our generator formula what else do i need i need n the number of turns that's
a hundred i need a and a clever way to handle this instead of saying pi r squared is to say well r
is d over two but when i square that i get a one quarter out in front so one quarter pi d
squared that just allows me to directly plug in the given quantity the advantage of that is if you
have to do precise error analysis that's really the right way to do it of course we're not doing
that in this video but i thought i'd mention it you're going to get .00785 square meters for that
a coil of that size is surely a very small number of square meters and then i need the
magnetic field strength which was given to me as half a tesla so i'm asked to find the peak voltage
and that's the coefficient of sine omega t in the generator formula we call it epsilon zero
and it's just the product of all these things so i have 100 times .00785 times half a tesla
times my angular speed so 52.4 radians per second and i get 20.6 volts for the peak induced voltage
now remember this is an ac voltage so i can also talk about the rms value of the
voltage and i'll post a link real quick back to where this was first discussed but as it turns
out the rms value that's a type of average is given by one over root 2 times the peak value
and that gives me 14.6 volts so we've created a simple 14.6 volt ac electric
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