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Induction II Law of Induction • The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is changing with time. d B | | dt d B | | N dt If coil has N turns B B dA Change in flux may be due to • Change in magnetic field • Change in the area • Both. Lenz’s law • The flux of the magnetic field due to the induced current opposes the change in the flux that causes the induced current. d B dt Motional EMF External agent pulls the loop with constant speed Induced current flows in the loop B BA B BDx d B | | dt | | BDv I ind | | BDv R R F1 is the net magnetic force • If external agent pulls with constant speed • Fext = F1 = Iind DB • Mechanical power P = F1 v The power expended by the external agent P F1v P I ind DBv 2 2 2 D Bv P R • A conducting rod of length L is being pulled along horizontal, frictionless and conducting rails. A uniform magnetic field fills the region in which the rod moves. Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s, resistance of rod as 415 m. •Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s resistance of rod as 415 m • Find Induced emf • = BLv = 0.619 V • Current in the conducting loop. • I = /R = 1.49 A •At what rate does the internal energy of rod increase? •P = Iind = 0.922 W •Force that must be applied by external agent to maintain its motion •F = ILB = 0.190 N •At what rate does this force do work on rod? •P = F v = 0.922 W Eddy Currents An emf and a current are induced in a circuit by a changing magnetic flux. When the magnetic flux through a large piece of conductor changes, induced current appear in the material in small loops. These are called eddy currents as they induce in little swirls/eddies. • http://www.ndted.org/EducationResources/HighSchool/Ele ctricity/eddycurrents.htm • http://www.ndted.org/TeachingResources/NDT_Tips/Lenz Law.htm Eddy currents and energy loss • They can increase internal energy and thus temperature of the material • Big eddy currents larger energy loss • Materials which are subjected to magnetic fields are often constructed in many small layers. Eddy currents slow down the motion of the conductor A cylindrical bar magnet is dropped down a vertical aluminum pipe of slightly large diameter . It takes several seconds to emerge at the bottom, whereas, identical piece of unmagnetized iron makes the trip in a fraction of a second. Explain why magnet falls more slowly?? Ans: delay is due to forces exerted on the magnet by induced eddy currents in the pipe. •Advantage Heating effect can be used in induction furnace. Magnetic field cannot force a stationary charge to move. Then why the charges move? Why there is an induced current? Induced electric fields A changing magnetic field induces an electric field. •Induced electric field exists, even when ring is removed. It is always tangential. Div E 0 Some facts • The driving force for induced currents is induced E-field • It exists, even when ring is removed. • It has no radial component. • As real as that might be setup by a real stationary charge. E ds d B E ds dt d E ds B da dt dB Curl E dt dB Curl E dt In the static case, Faraday’s law reduces to Curl E 0 E d s 0 You can not define a potential for an induced electric field. A uniform magnetic field B(t) pointing straight up fills the shaded circular region. If B is changing with time what is the induced electric field ? B(t) d E ds B da dt r d E ds dt B da d 2 E 2r B(t )r dt dB E 2r r dt 2 If B is increasing with time, induced current will run clockwise as look from above. A line charge is glued onto the rim of a wheel of radius R, which is then suspended horizontally . It is free to rotate. The spokes are made of wood. In the central region out to radius a there is a uniform magnetic field pointing up. Now someone turns the field off. What happens? B ds 2 dB E ds a dt Torque on the segment ds E ds R dB a R dt 2 Two parallel loops of wire are shown with common axis. Smaller loop is above the larger loop by a distance x>>R. Magnetic field due to current i in the larger loop is constant through the smaller loop and equal to the value on the axis. Suppose x is increasing with constant rate. (a) Determine the flux across the area bounded by smaller loop as a function of x. B 0 I 2 B R R 2 2 x 0 I R B BA 2 x 2 3/ 2 2 3 0 I R 2 x 2 3 r 2 Compute the emf generated in the smaller loop B BA 0 I R 2 x 2 3 r 2 d B 3 0 I R 2 r v 4 dt 2 x • Direction of current is anticlockwise as seen from above. 2 Two straight conducting rails form an angle where their ends are joined. A conducting bar in contact with the rails and forming an isoscale triangle with them, starts at the vertex at time t = 0 and moves with constant velocity v to the right. A magnetic field points out of the page. Find emf induced as a function of time. A x tan 2 2 B BA Bx tan 2 2 2 Bv t tan 2 2 A square loop of wire lies on a table, a distance s from a very long straight wire, which carries a current I. If someone pulls the loop away from the wire at speed v, what emf is generated? a a s a Flux through the loop a a s a B sa s 0 I ady 2y 0 Ia s a B ln 2 s 0 Ia s a B ln 2 s 0 Ia 1 ds 1 ds 2 s a dt s dt 0 Ia 1 v 2 s( s a) 2 • Direction of anticlockwise. induced current is