Mutual Inductance Suppose you have two coils with multiple turns close to each other, as shown in this cross-section We can define mutual inductance M 12 of coil 2 with respect to coil 1 as: Coil 1 Coil 2 B N1N1 N2N2 It can be shown that :
Inductors in Series What is the combined (equivalent) inductance of two inductors in series, as shown ? a b L2L2 L1L1 a b L eq Note: the induced EMF of two inductors now adds: Since: And:
Inductors in parallel What is the combined (equivalent) inductance of two inductors in parallel, as shown ? a b L2L2 L1L1 a b L eq Note: the induced EMF between points a and be is the same ! Also, it must be: We can define: And finally:
LC Circuits Consider the LC and RC series circuits shown: L C C R Suppose that the circuits are formed at t=0 with the capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why?? Consider from point of view of energy! In the RC circuit, any current developed will cause energy to be dissipated in the resistor. In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!
RC/LC Circuits RC: current decays exponentially C R i Q -i t 0 0 1 +++ - - - L C LC: current oscillates i 0 t i Q +++ - - -
LC Oscillations (qualitative) L C ++ - - L C L C ++ -- L C
Energy transfer in a resistanceless, nonradiating LC circuit. The capacitor has a charge Q max at t = 0, the instant at which the switch is closed. The mechanical analog of this circuit is a block–spring system.
LC Oscillations (quantitative) What do we need to do to turn our qualitative knowledge into quantitative knowledge? What is the frequency of the oscillations (when R=0)? –(it gets more complicated when R finite…and R is always finite) L C ++ - -
LC Oscillations (quantitative) Begin with the loop rule: Guess solution: (just harmonic oscillator!) where: determined from equation , Q 0 determined from initial conditions Procedure: differentiate above form for Q and substitute into loop equation to find . L C ++ - - i Q remember:
Review: LC Oscillations Guess solution: (just harmonic oscillator!) where: determined from equation , Q 0 determined from initial conditions L C ++ - - i Q which we could have determined from the mass on a spring result:
The energy in LC circuit conserved ! When the capacitor is fully charged: When the current is at maximum (I o ): At any time: The maximum energy stored in the capacitor and in the inductor are the same:
Lecture 22, ACT 1 At t=0 the capacitor has charge Q 0 ; the resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I . –What is the relation between 0 and 2, the frequency of oscillations when the initial charge = 2Q 0 ? (a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 2 0 1A
Lecture 22, ACT 1 At t=0 the capacitor has charge Q 0 ; the resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I . (a) I = I (b) I = 2 I (c) I = 4 I What is the relation between I and I , the maximum current in the circuit when the initial charge = 2Q 0 ? 1B
Summary of E&M J. C. Maxwell (~1860) summarized all of the work on electric and magnetic fields into four equations, all of which you now know. However, he realized that the equations of electricity & magnetism as then known (and now known by you) have an inconsistency related to the conservation of charge! I don’t expect you to see that these equations are inconsistent with conservation of charge, but you should see a lack of symmetry here! Gauss’ Law For Magnetism Faraday’s Law Ampere’s Law
Ampere’s Law is the Culprit! Gauss’ Law: Symmetry: both E and B obey the same kind of equation (the difference is that magnetic charge does not exist!) Ampere’s Law and Faraday’s Law: If Ampere’s Law were correct, the right hand side of Faraday’s Law should be equal to zero -- since no magnetic current. Therefore(?), maybe there is a problem with Ampere’s Law. In fact, Maxwell proposes a modification of Ampere’s Law by adding another term (the “displacement” current) to the right hand side of the equation! ie !
Displacement current Remember: EE I in I out changing electric flux
Maxwell’s Displacement Current Can we understand why this “displacement current” has the form it does? Consider applying Ampere’s Law to the current shown in the diagram. If the surface is chosen as 1, 2 or 4, the enclosed current = I If the surface is chosen as 3, the enclosed current = 0! (ie there is no current between the plates of the capacitor) Big Idea: The Electric field between the plates changes in time. “displacement current” I D = 0 (d E /dt) = the real current I in the wire. circuit
Maxwell’s Equations These equations describe all of Electricity and Magnetism. They are consistent with modern ideas such as relativity. They even describe light