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Walker, Chapter 23 Magnetic Flux and Faraday’s Law of Induction

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##### Walker, Chapter 23 Magnetic Flux and Faraday’s Law of Induction

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**Walker, Chapter 23Magnetic Flux and Faraday’s Law of**Induction Michael Faraday, portrait by Thomas Phillips**Induced EMF (Voltage) from changing Magnetic Flux**Electric currents produce magnetic fields. 19th century puzzle, can magnetic fields produce currents? A static magnet will produce no current in a stationary coil Faraday: If the magnetic field changes, or if the magnet and coil are in relative motion, there will be an induced EMF (and therefore current) in the coil. Key Concept: The magnetic flux through the coil must change, this will induce an EMF e in the coil, which produces a current I = e/ R in the coil. Such a current is said to be induced by the varying B-field.**Magnetic Flux**For a “loop” of wire (not necessarily circular) with area A, in an external magnetic field B,the magnetic flux is: q**Walker problem # 2**A uniform magnetic field of 0.0250 T points vertically upward. Find the magnitude of the magnetic flux through each of the five sides of the open-topped rectangular box shown in the figure, given that the dimensions of the box are L = 31.5 cm, W = 12.0 cm, and H = 10.0 cm.**Question: Magnetic Field from loop**• Current I flows around the loop as shown, driven by the battery. Inside the loop the magnetic field generated by current I is: • Out of screen • Into screen**** Magnetic flux generated by current in loop. • • • • In this case, we choose clockwise to be positive current. • In the plane of this picture, the magnetic field generated by this current is into the picture inside the loop, and out of the picture outside of the loop.**Examples of Induced Current**Any change of current in primary induces a current in secondary. The secondary shows a deflection only during the instant just after the switch is closed or opened.**Induction by Relative Motion**v • When a permanent magnet moves relative to a coil, the magnetic flux through the coil changes, inducing an EMF in the coil. • In a) the flux is increasing • In c) the flux is decreasing in magnitude. • In a) and c) the induced current has opposite sign. v**Faraday’s Law of InductionLenz’s Law**Faraday’s Law: The instantaneous EMF in a circuit (w/ N loops) equals the rate of change of magnetic flux through the circuit:**Question: Changing Magnetic Flux**• This is a plot of magnetic flux (Wb=T·m2) through a coil. • For which interval (a, b, or c) is the change in flux 0? c a b**Walker, Problem 23-9**• A 0.25 T magnetic field is perpendicular to a circular loop of wire with 50 turns and a radius 15 cm. • The magnetic field is reduced to zero in 0.12 s. • What is the magnitude of the induced EMF? (answer 7.35 V)**Inductance**• Consider a solenoid of length l with N windings and radius r (Area A=p r2). • A current I produces a magnetic field in the solenoid of B = m0 N I / l • This produces a total flux through each winding of F = A B = (m0 N A / l ) I • Define the Inductance L = (m0 N2 A / l ) (not a length!!) • If the current in the solenoid changes, there will be an induced EMF • e = - N DF/Dt = - (m0 N2 A / l ) (DI/Dt) • e = - L (DI/Dt)**Inductance**Walker Problem #42 Determine the inductance of a solenoid with 650 turns in a length of 25 cm. The circular cross section of the solenoid has a radius of 4.0 cm. [answer: 10.7 mH]**Example of Inductance**• Consider Solenoid from previous example, • Radius r = 5cm, length l = 10 cm, 3500 windings. • L = (m0 N2 Area / l ) = (m0 N2p r2 / l )**Transformers**• Microscopic currents in soft iron yoke amplify magnetic field produced by coils, and contain all magnetic field lines to “magnetic circuit” formed by iron. • Both primary and secondary circuits experience the same magnetic flux for each winding, but have different number of windings.**Flux in Transformer**F = flux through a single winding or either coil. (produced by currents in either or both circuits) EMF(primary) = - NPDF/Dt EMF(secondary) = - NSDF/Dt eP / eS = NP / NS Energy conservation requires eP IP = eS IS IP / IS = NS / NP**Walker problem #61**A step-up transformer has 20 turns on the primary coil and 500 turns on the secondary coil. If this transformer is to produce an output of 4600 V with a 12 mA current, what input current and voltage are needed? [answer: 0.3 A, 184 V]