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Transcript
PHYS152 Lecture 9 Ch 31 Induction and Inductance Eunil Won Korea University Fundamentals of Physics by Eunil Won, Korea University Induction: two experiments 1. A current appears only if there is relative motion between the loop and the magnet: the current disappears when the relative motion between them ceases 2. Faster motion produces a greater current 3. North pole toward the loop: clockwise current North pole away from the loop: counter clockwise current (For South pole, the situation is reversed) The current produced in the loop is called an induced current Two conducting loops close to each other but not touching 1. If we close switch S (to turn on a current in the right-hand loop), the meter suddenly and briefly registers a current - an induced current 2. If we open switch S (to turn on a current in the right-hand loop), the meter also registers a current (but in the opposite direction) Fundamentals of Physics by Eunil Won, Korea University Faraday’s Law of Induction An emf is induced in the loop at at left in figures when the number of magnetic field lines that pass through the loop is changing Magnetic flux : ΦB = ! ! · dA ! B SI unit of the magnetic flux: 1 weber = 1 Wb = 1 T m2 Faraday’s law becomes: dΦB E =− dt For the coil of N turns: dΦB E = −N dt Changing the magnetic flux: 1. Change the magnitude B 2. Change the area of the coil 3. Change the angle between the magnetic field and the area of the coil Fundamentals of Physics by Eunil Won, Korea University Lenz’s Law An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current Electric Guitar When the metal string is made to oscillate, it causes a variation in magnetic flux that induces a current in the coil Fundamentals of Physics by Eunil Won, Korea University Induction and Energy Transfers You must apply a constant force to pull the loop (magnetic force with opposite direction exists) The power need to pull: P = Fv The flux through the loop: ΦB = BA = BLx d dx dΦB = BLx = BL = BLv E= dt dt dt BLv i = E/R = R F2 and F3 cancel each other The force opposing you: F = F1 = iLB sin 900 = iLB BLv B 2 L2 v F = iLB = · LB = R R Let’s find the rate at which thermal energy appears in the loop Fundamentals of Physics by Eunil Won, Korea University P = i2 R ( rate of doing work: P = ! BLv R ! ×B ! F! = iL ) B 2 L2 v 2 P = Fv = R "2 B 2 L2 v 2 R= R : equal to the rate at which you are dong work on the loop Induced Electric Fields A changing magnetic field produces an electric field Reformulation of Faraday’s Law Consider a particle of charge q0 moving around the circular path The work done on it per one revolution by the induced electric field: Steady increase of B : counterclockwise current is induced An induced electric field exists even when the ring is removed Or, ! Eq0 F! · d!s = (q0 E)(2πr) so we find that E = 2πrE More generally, work done on a particle of charge q0 moving along any closed path: W = ! F! · d!s = q0 Now, we know that ! ! · d!s E dΦB E =− dt so we find that we get ! E= ! · d!s E dΦB ! E · d!s = − dt Observation: electric field lines due to a changing B field has a closed loop electric field lines due to an electric charge has no closed loop Fundamentals of Physics by Eunil Won, Korea University ! (Faraday’s law) Inductors and Inductance capacitor: produces a desired electric field inductor: produces a desired magnetic field (a long solenoid is a basic type of an inductor) N ΦB Inductance: L = i N: number of turns SI unit: 1 henry = 1 H = T m2/A Inductance of a solenoid (per unit length) N ΦB = nlBA the magnetic field of a solenoid is: B = µ0 in n: B: l: A: number of turns per unit length magnitude of the magnetic field length near the middle of solenoid cross sectional area N ΦB (nl)(BA) (nl)(µ0 in)(A) L= = = = µ0 n2 lA i i i L = µ0 n2 A l Fundamentals of Physics by Eunil Won, Korea University depends on the geometry of the device Self-Induction An induced emf appears in any coil in which the current is changing : this process is called self-induction (A self-induced emf will appear in the coil while the current is changing) For any inductor, Li = N ΦB If the current increases, the self-induced emf opposes the increase Faraday’s law tells us that d(N ΦB ) d(Li) di EL = − =− = −L dt dt dt (self induced emf) Fundamentals of Physics by Eunil Won, Korea University If the current decreases, the self-induced emf opposes the decrease RL Circuits When the switch is on: initially an inductor acts to oppose changes in the current through it. A long time later, it acts like ordinary connecting wire Applying the loop rule, di −iR − L + E = 0 dt or di E = Ri + L dt This form is mathematically identical to the case with the RC circuit. So we take the solution as: E i = (1 − e−Rt/L ) R or E i = (1 − e−t/τL ) R where L τL = R (inductive time constant) When the switch is off: di L + iR = 0 dt E −t/τL = i0 e−t/τL i= e R Fundamentals of Physics by Eunil Won, Korea University (decay of current) Energy Stored in a Magnetic Field Let’s rewrite the equation appeared in the RL circuit: dq Ei = E dt Ri2 di Li dt di E = Ri + L dt or di Ei = Ri + Li dt 2 : rate at which the emf device delivers energy to the rest of the circuit : rate at which energy appears as thermal energy : must be the energy per unit time stored in the magnetic field of the inductor Therefore, dUB = Li di di dUB = Li dt dt UB : the energy stored in the magnetic field Integrating this, we get: q2 UE = 2C 1 2 UB = Li 2 for the energy stored by a capacitor Energy Density of a Magnetic Field: in a solenoid, the energy density becomes 1 2 Li 2 2 2 2 UB L i µ0 n i µ0 n uB = = = = = Al Al l 2A 2 2 Fundamentals of Physics by Eunil Won, Korea University L = µ0 n2 A l 2 ! B µ0 n B = µ0 in "2 B2 = 2µ0 Note: we had 1 UE = !0 E 2 2 Mutual Induction We use the terminology mutual induction, when we hate two close-packed coils If the current in coil 1(2) changes, an emf will be induced in coil 2(1) We define the mutual inductance of coil 2 with respect to coil 1 as: M21 N2 Φ21 = i1 If the current varies with time, so dΦ21 di1 = N2 M21 dt dt The RHS is the emf appearing in coil 2: so M21 i1 = N2 Φ21 di1 E2 = −M21 dt dΦ21 E2 = −N2 dt If we interchange the roles of coils 1 and 2 di2 E1 = −M12 dt It is true (without proof) that Fundamentals of Physics by Eunil Won, Korea University M21 = M12 = M Summary Magnetic Flux Faraday’s law of Induction Emf and induced electric field Magnetic Energy Fundamentals of Physics by Eunil Won, Korea University ΦB = ! ! · dA ! B dΦB E =− dt E= ! ! · d!s E 1 2 UB = Li 2