February 23, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 25February 23, 2005 Physics for Scientists&Engineers 2 2InductionInduction! Last week we learned that a current-carrying loopin a magnetic field experiences a torque! If we start with a loop with no current in amagnetic field, and force the loop to rotate, wefind that a current is induced in the loop! Further, if we start with a loop with no currentand turn on a magnetic field without moving thecoil, again a current is induced in the loop! These effects are described by Faraday’s Law ofInduction and are the basis of electric motors andelectric power generation from mechanical motionFebruary 23, 2005 Physics for Scientists&Engineers 2 3FaradayFaraday’’s Experimentss Experiments! Consider the situation in which we have a wireloop connected to an ammeter so that we canmeasure current flowing in the loop! We hold a bar magnet some distance from theloop, pointing the north pole of the magnettoward the loop! While the magnet is stationary, there is nocurrent flowing in the loop! What happens if we move the magnet?February 23, 2005 Physics for Scientists&Engineers 2 4FaradayFaraday’’s Experiments (2)s Experiments (2)! When we move the magnet toward the loop, we induce apositive current in the loop! Now we turn the magnet around so that the south polepoints toward the loop! When we move the magnet toward the loop, we induce anegative current in the loopFebruary 23, 2005 Physics for Scientists&Engineers 2 5FaradayFaraday’’s Experiments (3)s Experiments (3)! Now let’s point the north pole toward the loop but moveaway from the loop• We get a negative current! We turn the magnet around so that the south pole pointstoward the loop and move away from the loop• We get a positive currentFebruary 23, 2005 Physics for Scientists&Engineers 2 6FaradayFaraday’’s Experiments (4)s Experiments (4)! We can create similar effects by placing a second loop near thefirst loop but with a more quantitative result as shown below! If a constant current is flowing through loop 1, no current will beinduced in loop 2! If we increase the current inthe loop 1, we observe that acurrent is induced in theloop 2 in the opposite direction! Thus not only does thechanging current in the firstloop induce a current in thefirst loop, the induced currentis in the opposite directionFebruary 23, 2005 Physics for Scientists&Engineers 2 7FaradayFaraday’’s Experiments (5)s Experiments (5)! Now if we have the current flowing in loop 1 in thesame direction as before, and decrease thecurrent as shown below, we induce a currentflowing loop 2 in the same direction as the currentin loop 1February 23, 2005 Physics for Scientists&Engineers 2 8Law of InductionLaw of Induction! From these observations we see that a changingmagnetic field induces a current in a loop! We can visualize the change in magnetic field as achange in the number of magnetic field linespassing through the loop! Faraday’s Law of Induction states that:•An emf is induced in a loop when the number of magneticfield lines passing through the loop changes with time! The rate of change of magnetic field linesdetermines the induced emfFebruary 23, 2005 Physics for Scientists&Engineers 2 9Magnetic FluxMagnetic Flux! To quantify the amount of magnetic field lines we definethe magnetic flux in analogy to the electric flux! When we introduced Gauss’ Law for the electric field, wedefined the electric flux as! For the magnetic field, we can define magnetic flux inanalogy as! The unit of magnetic flux is the weber (Wb) !E=!E • d!A" !B=!B • d!A"1 Wb = 1 Tm2February 23, 2005 Physics for Scientists&Engineers 2 10Magnetic Flux - Special CaseMagnetic Flux - Special Case! Consider the special case of a flat loop of area A in aconstant magnetic field B! In this case we can re-write the magnetic flux as•! is the angle between the surface normal vector of the plane of theloop and the magnetic field! If the magnetic field is perpendicular to the plane of theloop•! = 0°, !B = BA! If the magnetic field is parallel to the plane of the loop•! = 90°, !B = 0!B= BA cos"February 23, 2005 Physics for Scientists&Engineers 2 11FaradayFaraday’’s Laws Law of Inductionof Induction! We can then recast Faraday’s Law of Induction in terms ofthe magnetic flux as•The magnitude of the Vemf induced in a conducting loop is equal tothe time rate of change of the magnetic flux from the loop. Thisinduced emf tends to oppose the flux change.! Faraday’s Law of Induction is thus contained in the equation! We can change the magnetic flux in several ways includingchanging the magnitude of the magnetic field, changing thearea of the loop, or by changing the angle the loop withrespect to the magnetic fieldVemf= !d"BdtFebruary 23, 2005 Physics for Scientists&Engineers 2 12Induction in a Flat LoopInduction in a Flat Loop! Let’s explore Faraday’s Law for the special case of a flat loop inside amagnetic field that is constant in space, but that can vary in time! The magnetic flux for this case is! The induced emf is then! Carrying out the derivative we get! Taking d!/dt = " we get!B= BA cos"Vemf= !d"Bdt= !ddt(BAcos#)Vemf= ! A cos"dBdt! B cos"dAdt+ ABsin"d"dtVemf= ! A cos"dBdt! B cos"dAdt+#ABsin"February 23, 2005 Physics for Scientists&Engineers 2 13Induction in a Flat Loop - Special CasesInduction in a Flat Loop - Special Cases! If we leave two of the three variables (A,B,!) constant,then we can have the following three special cases• We leave the area of the loop and its orientation relative to themagnetic field constant, but vary the magnetic field in time• We leave the magnetic field as well as the orientation of the looprelative to the magnetic field constant, but change the area of theloop that is exposed to the magnetic field• We leave the magnetic field constant and keep the area of the loopfixed as well, but allow the angle between the two to change as afunction of timeA,! constant: Vemf= "A cos!dBdtB,! constant: Vemf= "B cos!dAdtA, B constant: Vemf=!ABsin"February 23, 2005 Physics for Scientists&Engineers 2 14Example: Changing Magnetic FieldExample: Changing Magnetic Field! A direct current of 600 mA is delivered to an idealsolenoid, resulting in a magnetic field of 0.025 T! Then the current is increased according to!Question:• If a circular loop of radius 3.4 cm with 200 windings islocated
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