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AC Circuits

AC Circuits
AC Circuits I Physics 2415 Lecture 22 Michael Fowler, UVa Today’s Topics • General form of Faraday’s Law • Self Inductance • Mutual Inductance • Energy in a Magnetic Field Faraday’s Law: General Form • A changing magnetic flux through a loop generates an emf around the loop which will drive a current. The emf can be written: d B E Ed  dt loop In fact, this electric field is there even without the wire: if an electron is circling in a magnetic field, and the field strength is increased, the electron accelerates, driven by the circling electric field—the basis of the betatron. The Betatron • If an electron is circling in a v • . magnetic field, and the magnetic field intensity is F increased, from Faraday’s law there will be circling lines of electric field which accelerate the electron. It is easy to design the field so that the electron circles at constant radius—electrons magnetic field perp into screen can attain 99.9 of the A betatron was used as a trigger in speed of light this way. an early nuclear bomb. V Clicker Question • You have a single loop of superconducting wire, with a current circulating. The current will go on forever if you keep it cold. • But you let it warm up: resistance sets in. • The current dies away, and therefore so does the magnetic field it produced. • Does this decaying magnetic field induce an emf in the loop itself A: Yes B: No. • (Assume there are no other loops, or magnets, etc., anywhere close.) V Clicker Answer • Does this decaying magnetic field induce an emf in the loop itself A: Yes B: No. • Yes it does The induced emf will be such as to produce some magnetic field to replace that which is disappearing—that is, in this case it will generate field going in through the loop, so the current will be as shown. • You could also say the induced emf is such as to oppose the change in current. • This is called “self inductance”. “Self Inductance” of a Solenoid • What emf E is generated in a • . solenoid with N turns, area A, for a rate of change of current dI/dt • Recall from Ampère’s law that B =  nI, so  =  NIA/ℓ. 0 B 0 • This flux goes through all N turns, so the total flux is N . B • Hence emf from changing I is: 2 N turns total in length ℓ: dNA dI B 0 EN . N/ℓ = n turns per meter. dt dtDefinition of Self Inductance • For any shape conductor, when the current changes there is an induced emf E opposing the change, and E is proportional to the rate of change of current. • The self inductance L is defined by: dI E L dt • and symbolized by: • Unit: for E in volts, I in amps L is in henrys (H). Mutual Inductance • We’ve already met mutual Coil 2: Coil 1: • . inductance: when the current N loops N loops 2 1 I in coil 1 changes, it gives 1 rise to an emf E in coil 2. 2 • The mutual inductance M is 21 M  N / I defined by: 21 2 21 1 where is the magnetic  21 flux through a single loop of Coil 2 coil 2 from current I in coil 1. 1 d dI 21 1 ENM 2 2 21 Coil 1 dt dtMutual Inductance Symmetry • Suppose we have two coils close to each other. A changing current in coil 1 gives an emf in coil 2: E M dI / dt 2 21 1 • Evidently we will also find: E M dI / dt 1 12 2 • Remarkably, it turns out that M = M 12 21 • This is by no means obvious, and in fact quite difficult to prove. Mutual Inductance and Self Inductance • For a system of two coils, such as a transformer, the mutual inductance is written as M. • Remember that for such a system, emf in one coil will be generated by changing currents in both coils, as well as possible emf supplied from outside. Energy Stored in an Inductance • If an increasing current I is flowing through an inductance L, the emf LdI/dt is opposing the current, so the source supplying the current is doing work at a rate ILdI/dt, so to raise the current from zero to I takes total work I 2 1 U LIdI LI 2  0 • This energy is stored in the inductor exactly 2 1 as is stored in a capacitor. U CV 2Energy Storage in a Solenoid • Recall from Ampère’s law that • . B =  nI, where n = N/ℓ. 0 • We found (ignoring end effects) the inductance 2 NA 0 L . • Therefore 2 2  NA B 22 0 1 1 1 LI A B /  0 2 2 2  N  0 2 1 an energy density B / inside. 0 2 Energy is Stored in Fields • When a capacitor is charged, an electric field is created. • The capacitor’s energy is stored in the field 2 1  E with energy density . 0 2 • When a current flows through an inductor, a magnetic field is created. • The inductor’s energy is stored in the field 2 1 B / with energy density . 2 0Mutual Inductance and Self Inductance • For a system of two coils, such as a transformer, the mutual inductance is written as M. • Remember that for such a system, emf in one coil will be generated by changing currents in both coils: dI dI 12 ELM 11 dt dt dI dI 12 EML 22 dt dtClicker Question • Two circular loops of wire, one small and one large, lie in a plane, and have the same center. • A current of 1 amp in the large loop generates a magnetic field having total flux  through S the small loop. • 1 amp in the small loop gives total flux  L through the large loop. A.  S L B.  S L C. =  S L Clicker Question • Two circular loops of wire, one small and one large, lie in a plane, and have the same center. • A current of 1 amp in the large loop generates a magnetic field having total flux  through S the small loop. • 1 amp in the small loop gives total flux  L through the large loop. A.  S L B.  S L C. =  M = M S L 12 21 Coaxial Cable Inductance • In a coaxial cable, the • . current goes one way in the central copper rod, the opposite way in the I enclosing copper pipe. • To find the inductance per unit length, remember the energy 2 1 Coaxial cables carry high frequency ac, U LI stored is in 2 such as TV signals. These currents flow the magnetic field. on the surfaces of the conductors. Coaxial Cable Inductance • To find the inductance • . per unit length, remember the energy 2 1 stored U  L I is in 2 I the magnetic field. • From Ampere’s Law the magnetic field strength at radius r (entirely from the inner current) B r is  I 0 B 2 rCoaxial Cable Inductance 2 1 • The energy stored U  L I 2 is in the magnetic field, • . 2 energy density . B /2 0 I • From Ampere’s Law, B I /2 r 0 so the energy/meter r 22 2 1 1 II 2 rdr r 22 00 2 LI B dv ln 22  2 2 8rr 4 o 1 r 1 B r  r 0 2 L ln from which the inductance/m 2 r 1
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