in this video we're talking about the motional emf induced across
a conductive body moving perpendicular to a magnetic field
and one way to look at this is in terms of the forces exerted on charges so in this rod i could
say there's electrons free to move and i want to think about the forces on these electrons
and i use the lorentz force right hand rule for this i put my fingers in the direction of v and
i sweep out to the direction of b my thumb points in the direction of the force on a positive charge
and i have to reverse that to get the force on a negative charge well that means the force on
these electrons is downward and as this rod cruises along in this magnetic field
you're going to end up with electrons accumulating at the bottom end of it and
that creates a negative potential there and you're going to end up with a deficit of electrons at the
positive end of it alternatively if you complete the circuit a current is going to end up flowing
electrons moving clockwise the current will be counterclockwise so this force analysis
tells us which way we would read the voltage with a voltmeter hooked up to this circuit
so this metal rod again is sliding along with the speed of v on these conductive wires and
my voltmeter is hooked up here to read the potential difference that's being induced
and that force argument has given me the answer of which way that goes and that's actually the
follow-up question at the end of our question so positive on this side negative on this side
now what we actually want to focus on in this video is making an electromagnetic induction
argument instead of a force on charges argument the argument that i just made about the force
on charges actually leads to exactly the same answer for the induced potential difference
and in fact it's the exact same physics as the hall effect and the hall voltage
so i'll post a link to that video real quick now we're taking an alternative approach to
it using electromagnetic induction and we're hoping the answer comes out to be the same
so instead of viewing it as forces on charges i look at this conductive loop that is growing
and i say as this loop grows as that metal bar slides along it's going to capture more and more
inward pointing magnetic field to oppose that change in flux my induced magnetic field
must point the opposite direction so increasing inward magnetic flux needs to be opposed by
induced outward magnetic flux and i ask which way does the current have to flow to get that done
and i use the ampere's law of right hand rule for this i line up my thumb in the direction of
current curl the fingers of my right hand around to create the desired flux and i get that leftward
in the top branch in other words counterclockwise would be the correct way for this current to flow
and if the circuit is broken by a voltmeter i'll have an accumulation of positive charge on this
end and negative charge on the other end and i get the same result for the sign of the voltage now
let's get a handle on the problem quantitatively by using this electromagnetic induction approach
so to do this i'm going to start by saying that this coil currently has a length of x and what i'm
trying to do here is express the flux through it so that i can use faraday's law so my initial flux
is going to be a b cosine theta well this loop is already perpendicular to the magnetic field
so the normal vector is parallel to it cosine theta i don't care about i know my flux is
maximized and it's going to be equal to a b and the area of my loop is given by x times l
and the strength of my magnetic field is b then i allow a short amount of time delta t to go by and
during that time the loop is going to move by an extra distance of delta x just a small increase my
loop is a little wider than it was before and my flux should be a little bigger than it was before
again it's just a b but it's my slightly larger a final and the width is now x plus delta x
my length stays the same and there's the strength of my magnetic field now i invoke faraday's law
and i'm really just concerned with the magnitude of the induced voltage i already figured out the
direction so i don't need the minus sign from faraday's law i only have one loop so n is one
and my delta phi the change in phi here is x plus delta x times lb minus my original
flux which was xlb and that's all divided by delta t when i look at the algebra in
the numerator i look at distributing the lb to both terms in this binomial
the first term would be an lb times x but i see a negative one of those
over here in the numerator so those are going to cancel and i get delta x times lb
over delta t and this is a very common trick in these derivations delta x over delta t appears
in this expression that's the speed so i end up with v l b now this is exactly the same result we
got for the hall voltage induced when the drift velocity was the speed that we used l was the
width of the conductor that we were using and b of course was the strength of the magnetic field
so it's just two different pathways in physical reasoning giving us the exact same answer
let's wrap things up by applying our new formula to a simple example so again the induced voltage
if i have a conductive body moving at right angles to a magnetic field is just going to be vl vl
b in this case we're talking about an airplane flying through a magnetic field perpendicular
to it and that's the earth's magnetic field we have a wing span of 28 meters
it's traveling 180 meters per second the wings are at right angles through the earth's magnetic
field so we don't have to worry about a cosine theta component and the strength of the field
is 0.4 gauss the gauss needs to be converted to tesla that's a factor of 10 to the negative 4.
other than that it's just a plug and chug problem i have a speed of 180
i have a length of 28 meters i have a magnetic field strength of 0.4 times 10 to the negative 4
tesla and i end up with a 0.202 volt induced potential difference between the tips of the wings
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