# chapt30 - Chapter 30: Induction and Inductance Problem 1 In...

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Chapter 30: Induction and Inductance Problem 1 In Fig. 30-37, the magnetic flux through the loop increases according to the relation Φ B = 6t 2 + 7t, where Φ B is in milliwebers and t is in seconds. (a) What is the magnitude of the emf induced in the loop when t = 2.0 s? (b) What is the direction of the current through R to the right or left? Answer (a) 31 mV, (b) to the left. B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 30-37 Solution (a) First, the “6” and “7” in the equations should have units. It should be 6 mWb/s 2 and 7 mWb/s. The magnitude of the induced emf is E = d Φ dt = 12 mWb s 2 t 7 mWb s At time t = 2.0 s E = d Φ dt = mWb s 2 2 s ( ) mWb s = 31 mV m (b) The direction of the induced current is in the direction that will oppose the change. If we view "the change" as an increase in the B field directed out of the page then an induced B field directed into the page would oppose it. Applying the right-hand-rule, the induced current would have to go around the circuit in a generally clockwise sense to produce a field into the page and the current would thus go from right to left in the resistor. Problem 4 An elastic conducting material is stretched into a circular loop of 12.0 cm radius. It is placed with its plane perpendicular to a uniform 0.800 T magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of 75.0 cm/s. What emf is induced in the loop at that instant? Answer 0.45 volt The magnitude of the induced emf is

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30-2 Chapter 32: Faraday’s Law of Induction E = d Φ dt = d BA ( ) dt = B dA dt = B d π r 2 ( ) dt = 2 Br dr dt E = 2 0.8 T ( ) 12 cm ( ) 75 cm s = 0.45 volt Problem 6 A uniform magnetic field B is perpendicular to the plane of a circular loop of diameter 10 cm formed from wire of diameter 2.5 mm and resistivity 1.69 × 10 -8 -m. At what rate must the magnitude of B change to induce a 10 A current in the loop? Answer 1.38 T/s Solution The current depends on the induced EMF, E , and the resistance, R, of the loop. i = E /R The induced EMF depends on the rate of change of magnetic flux inside the loop. If the diameter of the loop is D then E = d Φ B dt = A dB dt = D 2 4 dB dt The resistance depends on the geometry and resistivity of the wire that forms the loop. If the diameter of the wire is d then R = ρ l A = ρπ D d 2 2 = 4 D d 2 Combining these three equations and solving for the rate of change of B yields dB dt = 4 E D 2 = 4 iR D 2 = 16 i D D 2 d 2 = 16 i Dd 2 dB dt = 16 10 A 1.69 × 10 8 Ω⋅ m 10 cm 2.5 mm ( ) 2 = 1.38 T s
Chapter 30:Induction and Inductance 30-3 Problem 7 In Fig. 30-40, a wire forms a closed circular loop, with radius R = 2.0 m and resistance 4.0 . The circle is centered on a long straight wire; at time t = 0, the current in the long straight wire is 5.0 A rightward. Thereafter, the current changes according to i = 5.0 A – (2.0 A/s 2 )t 2 . (The straight wire is insulated, so there is no electrical contact between it and the wire of the loop.) What are the magnitude and direction of the current induced in the loop at times t > 0?

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## This note was uploaded on 08/03/2009 for the course PHYSICS 2049 taught by Professor C during the Spring '09 term at Vanderbilt.

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chapt30 - Chapter 30: Induction and Inductance Problem 1 In...

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