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Chapter 20: Induced Voltages and Inductances Induced Emf and Magnetic Flux Homework assignment : 17,18,57,25,34,66 Discovery of induction

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Magnetic flux A (area) magnetic flux: Induced Emf and Magnetic Flux

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Farady’s law of induction An emf in volts is induced in a circuit that is equal to the time rate of change of the total magnetic flux in webers threading (linking) the circuit: The flux through the circuit may be changed in several different ways 1)B may be made more intense. 2)The coil may be enlarged. 3)The coil may be moved into a region of stronger field. 4)The angle between the plane of the coil and B may change. Farady’s Law of Induction If the circuit contains N tightly wound loops

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Farady’s law of induction (cont’d) Farady’s Law of Induction The sum of E i s i along the loop is equal to the work done per unit charge, which is the emf of the circuit.

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Lenz’s law Farady’s Law of Induction The sign of the induced emf is such that it tries to produce a current that would create a magnetic flux to cancel (oppose) the original flux change. B due to induced current

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Lenz’s law (cont’d) The bar magnet moves towards loop. The flux through loop increases, and an emf induced in the loop produces current in the direction shown. B field due to induced current in the loop (indicated by the dashed lines) produces a flux opposing the increasing flux through the loop due to the motion of the magnet. Farady’s Law of Induction

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Motional Electromotive Force Origin of motional electromotive force I FBFB FEFE

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Motional Electromotive Force Origin of motional electromotive force I (cont’d)

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Motional Electromotive Force Origin of motional electromotive force II B. B ind

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Motional Electromotive Force Origin of motional electromotive force II (cont’d)

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Motional Electromotive Force Origin of motional electromotive force II (cont’d)

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Motional Electromotive Force Origin of motional electromotive force II (cont’d)

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Motional Electromotive Force Origin of motional electromotive force II (cont’d)

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Motional Electromotive Force Origin of motional electromotive force III

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Motional Electromotive Force Origin of motional electromotive force III (cont’d)

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Motional Electromotive Force Origin of motional electromotive force III (cont’d) JUST FOR FUN WITH CALCULUS!

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Motional Electromotive Force A bar magnet and a loop (again) In this example, a magnet is being pushed towards (away from) a closed loop. The number of field lines linking the loop is evidently increasing (decreasing).

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Motional Electromotive Force An electromagnet and a coil

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Motional Electromotive Force Tape recorder

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Generators A generator (alternator) The armature of the generator opposite is rotating in a uniform B field with angular velocity ω this can be treated as a simple case of the E = υ×B field. On the ends of the loop υ×B is perpendicular to the conductor so does not contribute to the emf. On the top υ×B is parallel to the conductor and has the value E = υB sin θ = ωRB sin ωt. The bottom conductor has the same value of E in the opposite direction but the same sense of circulation. top bottom v v B Farady emf

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Generators A generator (cont’d) AC generator DC generator

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Self-inductance Self-inductance Consider the loop at the right. - Switch closed : Current starts to flow on the loop. - Magnetic field produced in the area enclosed by the loop (B proportional to I). - Flux through the loop increases with I. - Emf induced to oppose the initial direction of the current flow. - Self-induction: changing the current through the loop inducing an opposing emf the loop. X X X X X X switch X X X X X X ab induced current

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Self-inductance Self-inductance (cont’d) I - The magnetic field induced by the current in the loop is proportional to the current: - The magnetic flux induced by the current in the loop is also proportional to the current: - Define the constant of proportion as L: self-inductance - From Frarady’s law: SI unit of L :

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Self Inductance Calculation of self inductance : A solenoid Accurate calculations of L are generally difficult. Often the answer depends even on the thickness of the wire, since B becomes strong close to a wire. In the important case of the solenoid, the first approximation result for L is quite easy to obtain: earlier we had Hence Then, So L is proportional to n 2 and the volume of the solenoid

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Self Inductance Calculation of self inductance: A solenoid (cont’d) Example: the L of a solenoid of length 10 cm, area 5 cm 2, with a total of 100 turns is L = 6.28×10 −5 H 0.5 mm diameter wire would achieve 100 turns in a single layer. Going to 10 layers would increase L by a factor of 100. Adding an iron or ferrite core would also increase L by about a factor of 100. The expression for L shows that μ 0 has units H/m, c.f, Tm/A obtained earlier

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RL Circuits Inductor A circuit element that has a large inductance, such as a closely wrapped coil of many turns, is called a inductor. RL circuit Kirchhoff’s rules: time constant

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Energy Stored in a Magnetic Field Inductor The emf induced by an inductor prevents a battery from establishing instantaneous current in a circuit. The battery has to do work to produce a current – this work can be considered as energy stored in the inductor in its magnetic field. energy stored in inductor

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