# Lec10 - Scott Hughes Massachusetts Institute of Technology...

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Scott Hughes 11 March 2004 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Lecture 10: Magnetic force; Magnetic fields; Ampere’s law 10.1 The Lorentz force law Until now, we have been concerned with electrostatics — the forces generated by and acting upon charges at rest. We now begin to consider how things change when charges are in motion 1 . A simple apparatus demonstrates that something wierd happens when charges are in motion: If we run currents next to one another in parallel, we Fnd that they are attracted when the currents run in the same direction; they are repulsed when the currents run in opposite directions. This is despite the fact the wires are completely neutral : if we put a stationary test charge near the wires, it feels no force. ±igure 1: Left: parallel currents attract. Right: Anti-parallel currents repel. ±urthermore, experiments show that the force is proportional to the current — double the current in one of the wires, and you double the force. 1 We will deviate a bit from Purcell’s approach at this point. In particular, we will defer our discussion of special relativity til next lecture.

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This all indicates a force that is proportional to the velocity of a moving charge; and, that points in a direction perpendicular to the velocity. These conditions are screaming for a force that depends on a cross product. What we say is that some kind of Feld ~ B — the “magnetic Feld” — arises from the current. (We’ll talk about this in detail very soon; for the being, just accept this.) The direction of this Feld is kind of odd: it wraps around the current in a circular fashion, with a direction that is deFned by the right-hand rule: We point our right thumb in the direction of the current, and our Fngers curl in the same sense as the magnetic Feld. With this sense of the magnetic Feld deFned, the force that arises when a charge moves through this Feld is given by ~ F = q ~v c × ~ B , where c is the speed of light. The appearance of c in this force law is a hint that special relativity plays an important role in these discussions. If we have both electric and magnetic Felds, the total force that acts on a charge is of course given by ~ F = q ˆ ~ E + ~v c × ~ B ! . This combined force law is known as the Lorentz force. 10.1.1 Units The magnetic force law we’ve given is of course in cgs units, in keeping with Purcell’s system. The magnetic force equation itself takes a slightly di±erent form in SI units: we do not include the factor of 1 /c , instead writing the force ~ F = q~v × ~ B .
This is a very important diference! It makes comparing magnetic efects between SI and cgs units slightly nasty. Notice that, in cgs units, the magnetic Feld has the same overall dimension as the electric Feld: ~v and c are in the same units, so ~ B must be ±orce/charge. ²or some odd reason, this combination is given a special name: 1 dyne/esu equals 1 Gauss (1 G) when the ±orce in question is magnetic.

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## This note was uploaded on 04/11/2008 for the course PHYSICS 8.022 taught by Professor Hughes during the Spring '08 term at MIT.

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Lec10 - Scott Hughes Massachusetts Institute of Technology...

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