chapter29---PHY 131 JAB - Chapter 29 Electromagnetic...

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Chapter 29: Electromagnetic induction
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§29.1-3: Faraday’s Law and Lenz’s Law dt d N B Φ - = ε An induced emf in a “coil” of N loops is due to a changing magnetic flux. Ways to induce an emf: 1. Vary the magnetic field 2. Vary the area of the coil 3. Change the angle between B and A Faraday’s law:
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The magnetic flux is proportional to the number of B-field lines that cross a given area. Loop of wire with area A θ cos BA B = Φ The unit of magnetic flux is the weber: 1 Wb = 1 Tm 2
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The direction of induced emfs and currents always oppose the change in flux that produced them. That is, the induced I (and thus induced B) tries to keep the total flux through the loop constant. Lenz’s Law
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Example: A long straight wire carrying a steady current is in the plane of a circular loop of wire. If the loop of wire is moved closer to the wire, what is the direction of the induced current in the wire loop? I Wire loop There is a magnetic field into the page at the location of the loop. As the loop gets closer to the wire there is an increase in flux. To negate this increase in flux, the induced B-field must point out of the page. This requires a CCW current.
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Example: If the magnetic field in a region varies with time according to the graph shown below, find the magnitude of the induced emf in a single loop of wire during the following time intervals (a) 0-2.0 ms, (b) 2.0-4.0 ms, and (c) 4.0-8.0 ms. The loop has area 0.500 m 2 and the plane of the loop is perpendicular to the B-field. 0.50 T 2 4 8 B (T) t (ms)
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Δ Δ - = Δ ΔΦ - = t B A t B ε Using Faraday’s Law: This is the slope of the given B versus time graph. Example continued
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Example continued ( ) V 130 s 10 0 . 2 T 00 0 T 50 . 0 m 500 . 0 3 2 = × = Δ Δ - = - . - t B A ε (a) In the interval 0.0-2.0 ms: ( ) V 0 s 10 0 . 2 T 50 0 T 50 . 0 m 500 . 0 3 2 = × = Δ Δ - = - . - t B A ε (b) In the interval 2.0-4.0 ms:
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( ) V 63 s 10 0 . 4 T 50 0 T 00 . 0 m 500 . 0 3 2 = × = Δ Δ - = - . - t B A ε (c) In the interval 4.0-8.0 ms: Example continued
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Example continued (d) What is the direction of the induced current in parts a, b, & c? Take the positive direction to be out of the page. For a, the flux is increasing, so B ind must be into the page and I ind is CW. For b, the flux is constant, so B ind = 0 and I ind = 0 For c, the flux is decreasing, so B ind must be out of the page and I ind is CCW.
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Example (29.7): A closely wound coil with N turns of cross-sectional area A lies in the xy-plane. The coil is placed in a uniform but time dependent magnetic field that is parallel to the z-axis.
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