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PHYSICS 2B - Lecture Notes
Ch. 31: Electromagnetic Induction
Preliminaries
had little formal education and no training in mathematics, but he had remarkabl^ Michael Faraday (1791-1867)^ of Britain is regarded as one of the greatest of experimenters.e physical intuition. He^ He originated the concept of a field and is credited with inventing the electric motor and the dynamo. knew that an electric field in a conductor gives rise to an electric current, which in turn produces a He magnetic field. field. We now consider the process called He reasoned that it must also be possible to produce an induction. electric field from a magnetic
Faraday’s Law
the north pole^ A^ very dramatic lecture demonstration consists of a solenoid connected to a galvanometer. As of a magnet is inserted into the solenoid, the galvanometer will deflect, say, to the left. With the magnet inserted, the galvanometer reading returns to zero. galvanometer will deflect to the right. If the magnet is now reversed and the south If the magnet is withdrawn, the pole inserted, the deflection will be to the right and to the left on withdrawal. to be induced. Faraday reasoned that the induced emf in the galvanometer circuit was related to the The current through the galvanometer is said change in magnetic flux through the circuit. Recall, the magnetic flux through an area A is,
! B = # A B r " dA r.
Faraday inferred that the results of his observations could be summarized as,
E = " d d^! tB.
This is Faraday’s law of induction and is the integral form of the fourth of Maxwell’s equations.
is two-sided, there is an ambiguity in the choice of direction of the normal to the ar^ It is important to remember the directionality of the^ integrals in this expression. Since the areaea. However, once^ A the normal direction is chosen, the positive sense for the integral implied in the emf is given by the right hand screw rule. The negative sign above indicates, that if the area lies in the plane of the page and the - normal direction is ch clockwise, so that if the flux integral is positive, the induced emf would drive a current in theosen to be out of the page, the positive sense of the line integral would be counter clock-wise- direction. It is interesting to consider ways in which the magnetic flux integral might change:
- the magnitude of the magnetic field can changethe magnitude of the area can change
- the relative orientation of the magnetic field and the area can change.
Any of these can give rise to an induced emf and examples of each are given in your text.
Lenz’s Law
out of the page^ Consider. A magnetic field is^ wire loop of^ area^ A applied also pointing out of the page so that the magnetic flux is^ in the plane of the paper with^ the^ normal^ direction taken as pointing positive. By Faraday’s law Allow the magnetic field to increase with time so that the time derivative of the flux is positive. the induced emf is clockwise as is the induced current in the wire. This induced current produces a magnetic field of its own that points into the page. That is, the induced current produces a magnetic field that opposes the change in the applied field. This statement is called thought of as an expression of the Lenz’s Law Conservation of Energy. and is represented by the negative sign in Faraday’s law If Lenz’s law did not hold, the induced field_._ It may be could add to the applied field so that the total field could increase without limit. Non-Conservative Electric Fields We saw that the electric field produced by a static array of charges is conservative, that is
C "^ E r^^!^ d rl=^0 ,
for any closed path Faraday’s law as C. A different situation arises, however, in the case of induced emfs. We write B C A E d d^ d B dA dt dt
E = $ r^ " r l^ = #! = # $r" r.
Clearly, if there is a changing magnetic flux thro will not be conservative. ugh the area encircled by C , the induced electric field
Betatron and Tokamak
which was used to accelerate electrons.^ This principle is the basis of the operation of an early particle accelerator called the A strong magnetic field was applied perpendicular to the plane^ betatron of a torus. The induced electric field inside the torus served to accelerate a beam of electrons. arrangement provides part of the heating of the plasma confined in a tokamak. This is a toroidal A similar device to induce nuclear fusion as potential source of energy. Superconductivity and Perfect Diamagnetism
exactly zero below a critical temperature. This is ca^ At low temperatures some materials have the property that their electrical resistance becomeslled superconductivity and is a totally quantum mechanical phenomenon a current is induced in a superconduct that cannot be explained classically. As a result of having zero resistance, onceor, it persists indefinitely; in some cases, for years. When a magnet is brought close to a superconductor, it induces eddy currents that produce a magnetic field that exactly cancels the applied field, so that the magnetic field inside the superconductor remains exactly zero. This is called the the material is Meissner effect - 1. and is an example of perfect diamagnetism for which the susceptibility of
