Note 9: Faraday s Law
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1 Note 9: Faraday s Law In 1831, Faraday discovered that EMF (electromotive force, i.e., voltage) was induced by time varying magnetic flux. This was a monumental discovery in the physics history. Before Faraday, various attempts had been made to produce electric field from magnetic filed since it was known that magnetic field was produced by electric field. Power generation at large power plants and wind power generation all depends on the Faraday's law. Photocell produces electricity directly from solar radiation. This is quantum mechanical effect.
2 (a) North pole of bar magnet is approaching a coil. The magnetic flux in the coil is increasing with time. The current driven by the induced EMF tends to oppose the increasing ma gnetic flux. ( b) No motion no EMF (c) When the north pole is moving away from the loop, EMF opposite to the case (a) is induced. The direction of the current is reversed.
3 EMF induced by Time Varying Magnetic Flux Faraday's law can be expressed as Electromotive force = EMF d where B d A is the magentic flux (Wb). A Change in the magnetic flux is due to either timevarying B magnetic field, change in the area t the angle between B and A. The magnetic flux through an area A is = AB cos (Wb) where is the angle between the B field and the vector normal to A. Change in the flux can be due to that in B, A, or. da, or change in
4 Example 1. When a magnetic field normal to a square coil of 18 cm having 00 winding turns increases from B 0 to 0.5 T in 0.8 sec, what is the induced EMF? Sol. The magnetic flux linked to the coil is T NABcos (m ) t t (Wb) 0.8 sec d EMF 4.0 V. Note cos 0 1. The minus sign is w.r.t. the direction of the current to produce the B field. Example. A rectangular coil of area A ab is rotating in a magnetic field B at angular frequency rad/sec. Find the EMF to be induced. Sol. The magnetic flux through the coil is changing with time as = abb cos t (Wb) where t 0 is chosen when the coild plane is normal to B (maximum flux). d Induced EMF is EMF ( abb cos t) abb sin t This is the principle of ac generator.
5 Example 3. A conductor loop of radius a ia approaching a current carrying wire at a velocity v. Determine the direction of current induced in the loop. Sol. The current is the loop will flow so as to oppose the change in the magnetic flux enclosd by the loop. The magnetic flux out of the page increases as the loop approaches the wire. Therefore, the current will flow clock wise. If the coil is moving away from the current, the EMF is reversed. v I I'
6 Example 4. A rectangular loop of area ab is moving away from a current carrying wire at velocity v. Find the EMF to be indued. Sol. The magnetic field due to the current carrying wire is 0I B r The magntic flux through the loop is 0Ib vta 1 0Ib vt a = dr ln vt r vt Then the induced EMF is d 0Ib a 1 EMF a vt t The secondary current I ' in the loop would flow clockwise. (Its magnitude depends on the resistance in the loop.) I I ' b a v vt a + vt
7 Motional EMF In the figure, a conducting bar is moving across the magentic field at a vlocity v. The magnetic flux lxb lvtb is increasing and the induced EMF is EMF lvb (V) The minus sign indicates that the current fill flow so as to oppose the change in the magnetic flux, i.e., counterclockwise in this case. This is known as Lenz's law.
8 Example 5. Electromagnetic Braking. A conducting bar of mass m moves across a magnetic field B at an initial velocity v on frictionless rails. The total resistance of the circuit is R. When the external force is removed at t 0, show that the velocity decreases expoenentially with time. i Sol. The EMF is Blv( t) and the current I can be found from Blv t RI 0 The equation of motion for the bar gives dv lb m IlB v R t mr Solution is v t vi exp, (sec) lb
9 If the velocity is reversed, so is EMF. Motional Electric Field: E = v x B (V/m)
10 Example 6. A conductor rod of length l is rotating in a magnetic field at an agular velocity (rad/sec). Calculate the EMF induced. Sol. At radius r, the electric field is E vb rb r Then, the EMF is (V/m) l 1 EMF Erdr Bl 0 (V) If a conducting disk of radius a is rotating in a magnetic field, an EMF is induced between the center and the edge, 1 EMF Ba (V) This is the principle of homopolar generator.
11 Example 7. Problem 5. B 0.5T, l 1. m, R 1, 5, m 0.5kg. Find the terminal velocity v. lbv cos Sol. EMF = lbv cos RI I R The source of motion is the gravitational potential energy which is converted to heat in the resistor R. d RI mgh mgv sin lbv cos R mgv sin R Rmg sin sin 5 v.8 m/s lb cos cos 5
12 Example 7. Faraday tried to measure motional emf across the River Thames. The sensitivity of instruments was not high enough and he could not confirm. 4 If the earth magentic field (nearly downward) is 5 10 T and water flow velocity is 3 m/sec, what emf is expected across the river wih of 150 m? 4 Ans. emf = vbl V Example. Having learned that an object moving across a magnetic field should induce an emf, a student tried to measure it in a car which goes practically normal to the earth magnetic field. He used a volt meter to measure voltage across the wih of the car. Explain why he failed. Ans. The magnetic flux enclosed by the car is constant and there should be no emf.
13 Example 9. The current of a long solenoid having winding density n (turns/m) is given by I cos t. Find the electric field at distance r from the axis of the solenoid. Sol. The magnetic field in hte solenoid is B ni cos t. Outside the sollenoid r R d re EMF R 0nI0 cos t R 0I0 sint 1 R E 0I0sint r Inside r R d re EMF R 0nI0 cos t R 0I0 sint 1 E r0i0sint
14 Generators and Motors AC Generator When a coudctor loop rotates in a magetic field, a sinusoidal voltage is induced. The magnetic flux is = AB cos t. Induced EMF is d EMF ABsin t
15 DC Generator If a commutator is installed, the voltage becomes unipolar. Practical dc generator for smooth dc EMF
16 (A/m ) Eddy Current In a conductor, electric field drives current through the Ohm's law J E where (S/m) is the conductivity. When a conductor plate is placed near a coil carrying time varying current, electric field is induced and electric current is driven. The current heats the conductor (Joule heating). Induction kitchen stove uses this principle. In the pendulum shown, the one without slits damps faster because eddy current can flow without being impeded. The slits prevents eddy current and oscillation lasts longer.
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