in this video we're given a rather large coil it has 10 000 turns a diameter of 50 centimeters
and it's rotating from the coil orientation being parallel to the magnetic field to being
perpendicular to the magnetic field and what we're using here to induce an emf is actually
just the magnetic field of the earth which is rather weak so i'm saying it's half a gauss here
and we do this 90 degree turn in 0.8 seconds so i can see that in the initial state the magnetic
flux is 0 and that's because the plane of each of the loops in this coil is parallel to the magnetic
field so there are no magnetic field lines penetrating the loops there's no magnetic flux
in the second picture i have maximum magnetic flux every turn in that coil is
perpendicular to the magnetic field and this maximizes the number of captured field lines
so you might remember that the flux formula is given by a b cosine theta where that angle is
measured between a normal vector for the coil and the direction of the magnetic field and that angle
is going to be either 0 or 180 it doesn't matter because we're not concerned about sines here
so the cosine part turns out to be a and 1. end up with a magnetic flux in the final state
and this is just the flux for one turn of the coil given by a times b so i need the area of a loop
you can use pi r squared or you could use one quarter pi diameter squared
same thing times the strength of the magnetic field and i plug in the numbers that's one
quarter pi times .5 squared times the strength of the magnetic field now this needs to be
expressed in tesla and the conversion is that a gauss is 10 to the negative fourth
tesla so we can turn this magnetic field into 5 times 10 to the negative fifth
tesla and i get a flux per loop of 9.82 times 10 to the negative 6 tesla meters squared now that i
have the change in the flux for this rotation i can invoke faraday's law and i'm just going
to write this for the magnitude of the induced emf because we don't care about the direction
that's going to be n that's the number of loops that's where it comes into the calculation
times the rate of change in the magnetic flux through one loop and we have 10 000 loops
and we have a change in flux that's final minus initial
because the initial was zero this is simple it's just the final value over the amount of time it
took to change that flux that was 0.8 seconds and i get 0.123 volts induced by rotating this coil
in part b we're asked to compute the necessary sustained rotation rate of the coil to produce an
rms voltage or emf the way i set it here of 1.0 volts by using the earth's magnetic field
so now we're going to appeal to the generator formula and i'll post a link real quick to where
this was derived and i'm thinking about the sustained voltage that's produced by
continuing this rotation over time and the generator formula said this the voltage as
a function of time produced by a rotating loop in a magnetic field is given by some peak value
times the sine of omega t where omega is the angular velocity in radians per second and
it turned out that peak value is given by n a b times omega now in this case we want to get
an rms or root mean squared voltage of one volt and we have to relate that to the peak
voltage so i'll post a link real quick to the video where this was first introduced when we were
first looking at ac circuits and the relationship is that the rms value is 1 over root 2
times the peak value since we're looking at an rms value of 1 volt in this case it means
our peak value is root 2 times 1 volt or root 2 volts so that's what we're going to plug in for
the peak voltage in the generator formula so i'm going to go ahead and solve this formula for omega
omega is our peak voltage divided by n a b our peak voltage is given by root two volts
n is the number of turns that's ten thousand a
again is one quarter pi times diameter squared so 0.5 squared
and b is 5 times 10 to the negative fifth tesla when i run the numbers on this i get 14.4 radians
per second and that's a rotation rate but i want to convert that to rotations per minute which is a
more standard way of talking about how fast things are spinning so i take my 14.4 radians per second
i multiply by the conversion factor 2 pi radians in the denominator for every 1 rotation
and another one 60 seconds in the numerator for every 1 minute in the denominator cancel
my radians cancel my seconds and when i run the numbers on this to three sig figs i get 138
rotations per minute so as i pointed out in the video for the generator formula
this idea of a coil rotating in a magnetic field is the basis of how we generate
electrical power be true for a coil driven by a windmill or a steam driven turbine
in the gas powered or nuclear powered power plants or a coil driven by the turbine in a
hydroelectric dam you just put an axle through one of these things use some kind of source of
mechanical energy to spin it around and you're automatically generating alternating currents
in this case all you have to do is take this relatively small coil and spin it a little faster
than two revolutions per second and you're gonna read an rms value of one volt at the terminals
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