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**Unformatted text preview: **homework 09 – KIM, JI – Due: Mar 23 2008, 4:00 am 1 Question 1, chap 31, sect 2. part 1 of 1 10 points A square loop of wire of resistance R and side a is oriented with its plane perpendicular to a magnetic field vector B , as shown in the figure. B B a I What must be the rate of change of the magnetic field in order to produce a current I in the loop? 1. dB dt = I a 2 R 2. dB dt = Ra I 3. dB dt = I Ra 4. dB dt = I R a 2 correct 5. dB dt = I a R Explanation: The emf produced is given by E = dφ dt = a 2 dB dt . Using Ohm’s law we can solve for B I R = a 2 dB dt dB dt = I R a 2 . Question 2, chap 31, sect 2. part 1 of 1 10 points A toroid having a rectangular cross section ( a = 2 . 62 cm by b = 4 . 11 cm) and inner ra- dius 2 . 34 cm consists of N = 210 turns of wire that carries a current I = I sin ω t , with I = 31 . 6 A and a frequency f = 42 . 9 Hz. A loop that consists of N ℓ = 29 turns of wire links the toroid, as in the figure. b a N R N l Determine the maximum E induced in the loop by the changing current I . Correct answer: 0 . 275601 V (tolerance ± 1 %). Explanation: Basic Concept: Faraday’s Law E = − d Φ B dt . Magnetic field in a toroid B = μ N I 2 π r . Solution: In a toroid, all the flux is confined to the inside of the toroid B = μ N I 2 π r . So, the flux through the loop of wire is Φ B 1 = integraldisplay B dA = μ N I 2 π sin( ω t ) integraldisplay b + R R adr r = μ N I 2 π a sin( ω t ) ln parenleftbigg b + R R parenrightbigg . Applying Faraday’s law, the induced emf can be calculated as follows E = − N ℓ d Φ B 1 dt = − N ℓ μ N I 2 π ω a ln parenleftbigg b + R R parenrightbigg cos( ω t ) = −E cos( ω t ) homework 09 – KIM, JI – Due: Mar 23 2008, 4:00 am 2 where ω = 2 πf was used. The maximum magnitude of the induced emf , E , is the coefficient in front of cos( ω t ). E = − N ℓ d Φ B 1 dt = − N ℓ μ N I ω 2 π a ln bracketleftbigg b + R R bracketrightbigg = − (29 turns) μ (210 turns) × (31 . 6 A) (42 . 9 Hz) (2 . 62 cm) × ln bracketleftbigg (4 . 11 cm) + (2 . 34 cm) (2 . 34 cm) bracketrightbigg = − . 275601 V |E| = 0 . 275601 V . Question 3, chap 31, sect 1. part 1 of 2 10 points In the arrangement shown in the figure, the resistor is 2 Ω and a 2 T magnetic field is directed out of the paper. The separation between the rails is 4 m . Neglect the mass of the bar. An applied force moves the bar to the left at a constant speed of 4 m / s . Assume the bar and rails have negligible resistance and friction. m ≪ 1g 4 m / s 2Ω 2 T 2 T I 4m Calculate the applied force required to move the bar to the left at a constant speed of 4 m / s. Correct answer: 128 N (tolerance ± 1 %)....

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