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lec14

# lec14 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 31 March 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 14: Induction: Faraday’s & Lenz’s laws 14.1 Moving wires in magnetic fields 14.1.1 Single length Suppose we take a length of conducting wire and drag it through a uniform magnetic field: B v F exerted on positive charges Charges in the wire will feel a force due to being dragged along through the vector B -field. This leads to a separation of charges in this rod: as drawn, the left end of the rod will acquire a net + charge, the right end acquires a net - charge. How much charge separation occurs? By moving the charges apart, we must create an electric field. The electric field will grow until the force that it exerts on the charges balances the magnetic force: q vector E balance + q vectorv c × vector B = 0 -→ E balance = - vectorv c × vector B . 127

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14.1.2 Closed loop Suppose we now take our wire and bend it into a square. What happens when we drag it through the magnetic field? Well, if the magnetic field is truly uniform and filling all of space, nothing very interesting happens: B uniformly fills plane v + - - + We get charge separation in both the top and the bottom legs, so there’s an vector E -field pointing to the right along both legs. This stops the flow of charge — once the vector E -field is built up, the charge separation doesn’t do anything else. It’s done. Things get more exciting if we make the magnetic field non-uniform. Suppose that the magnetic field is zero below some line, but is constant and non-zero above that line: v + - uniform dotted line above B Because of the magnetic force, charge will flow — as we already know — to the left across the upper side of the loop. However, there’s now no opposing vector E -field that will prevent it from continuing to flow! The charges will very happily circumnavigate the entire loop, forming a cyclical current. We say that we have induced a current to flow in this loop. This is our first encounter with the phenomenon of electromagnetic induction. A few other things are worth noticing. First, note that contintegraldisplay wire loop vector E · dvectors negationslash = 0 . For this to be true, we cannot have vector ∇ × vector E = 0! Don’t be too disturbed by this — all this tells us is that we have begun to move beyond electrostatics. 128
Second, notice that the current flowing in this loop will itself feel a force from the magnetic field: the leg across the top of the loop carries a current to the left. Doing the cross product for vector I × vector B , we find that this generates a force on the loop that points up — opposing the motion of the loop. (There is also a force to the left on the left-hand side of the loop, and a force to the right on the right-hand side, but these are equal and opposite. The force on the top of the loop is unopposed.) This opposition of the loop’s motion is our first contact with a more general rule called Lenz’s law , which is a kind of electromagnetic inertia.

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