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## Chapter 32 Inductance

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**Chapter 32 Inductance**PHYS 2326-21**Concepts to Know**• Self Induction • Mutual Inductance • Inductors • Magnetic Field Energy • RL Circuit • LC Circuit • LRC Circuit**Self Induction**• Must distinguish between emf from sources like a battery and induced emf from changing magnetic fields**Mutual Inductance**• Magnetic flux variation in one circuit can cause a magnetic flux variation in another circuit. Note that this can be by intention or by accident. • By intention, one can have a transformer • By accident, one relay might cause another relay to close or open. or noise to be injected in one circuit by another**Mutual Inductance**• emf2 = -N2 dΦ12/dt = -N2 d(M12 I1/N2)/dt = • -M12 dI1/dt , so emf1 = -M21 dI2/dt • Note that also M12 = M21 = M • That is a current in coil 1 generates current in coil 2 and a current in coil 2 generates a current in coil 1**Inductor**An inductor is a circuit element that primarily has large inductance It can be a solenoid valve, or relay or even merely a component intended to provide inductance to a circuit for some purpose Inductors are subject to Lenz’s law which states that with a changing magnetic field, there will be an action taken to oppose that change**Magnetic Field Energy**• Given an applied emf across an inductor in series with a resistance,**Transformer**• A transformer is two coils with total mutual induction Read Chapter 33.8**Permeability**Ampere’s law applied to a toroid Note that the magnetic field B depends upon μo the permeability of free space. This is for the area in the torus that the winding goes around What happens if it’s not a vacuum? Most transformers and solenoids have metal cores in the coils**Permeability**• Note this doesn’t seem to even be in this text book In the same concept as with dielectrics for capacitors there is a magnetic equivalent to the dielectric constant Km = B/Bo = relative permeability Note though that K for dielectrics ranged from 1 for a vacuum upwards to ?? Km is 1 for a vacuum, > 1 for paramagnetic materials, slightly smaller than 1 for diagmagnetic materials and >> 1 for ferromagnetic materials**Permeability**μ = μo Km is the permeability of a material**Example 1**Tesla coil has N1 turns along length l of a hollow tube and a second coil of N2 what is a) the mutual inductance b) if N1=10,000 and N2 = 100, l = 1m, radius 1cm what is the value of M? c) if a radio frequency of 1000 KHz is sent through coil 2 so that current oscillates with amplitude of 100ma what is the average magnetic flux through coil1 d)max current through coil1 e)max induced emf in coil1 f)back emf in coil 1?**Example 1**• M = N2ΦB2 /I1, ΦB2 = ΦB1 =B1A A = πr2 = 3.14(0.01)^2 = 0.000314 m2 B1=μo N1 I1/l substituting for B1 M = (μo N1 I1/l ) N2A / I1 = μoN2N1 A I1 / l b) M = 3.946 E -4 H c) max flux? I2 = Iosin ωt , ω = 2 πf (angular freq.)=6.28E+6 rearranging M, ΦB2 = ΦB1 = M I2 / N2 = (3.946E-4)(0.1)/(100) = 3.948E-7 Wb**Example 1**d) coil 1 current I1 = I2 N2/N1 = (0.1)(100)/(10,000) = 1.0 E-3A e) emf1 = -M dI2 /dt , I2 = Io sin ωt dI2 /dt = Ioω cos ωt emf1 = -M Ioω cos ωt emf1max = M Ioω = (3.948E-4)(0.1)(6.28E+6) = 248 V f) emf2max = emf1max N2/N1 =(248)(100/10,000) = 2.48 V**Example 2**a) Inductance of a long solenoid length l and area A with N turns? b)if 2m long 2cm radius and 2000 turns? c) if current decreased from 4A to 0 in 2 microseconds what is magnitude and direction of the self induced emf ? d) what is the energy stored in the solenoid at the beginning of the 2 microsecond interval? e) How much electrical power is dissipated during this time?**Example 2**• inductance Substituting for B in L**b) Inductance value**A = 1.257E-3 L = (1.2566E-6)(2000)^2(1.257E-3)/2.0 = = 3.159E-3 H c) emf? emf = (3.159E-3)(4-0)/(2.0E-6) = 6318V in direction of current trying to stop field collapse by trying to maintain current**d) Energy?**U1 = (1/2) (3.158E-3)(4)^2 = 2.52E-2 Joules e) Power P = (2.52E-2)/(2E-6) = 12,632 W why so high? It’s also about timing too**RL Circuits**• Review 32.2**LC Circuits**Study Chapter 32.5 Given a capacitor and an inductor at time t=0 with the capacitor being connected in series to the inductor with Qmax charge on it there will be an oscillation. If no resistance is there to dissipate the energy, it will continue to oscillate**RLC Circuit**Study Chapter 32.6 The difference between an LC and RLC circuit – other than resistance exists in all circuits is that R is a dissipative element that absorbs energy as current flows