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Chapter 30 Inductance. Magnetic Effects. As we have seen previously, changes in the magnetic flux due to one circuit can effect what goes on in other circuits The changing magnetic flux induces an emf in the second circuit. Mutual Inductance. Suppose that we have two coils,

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Chapter 30InductanceMagnetic EffectsAs we have seen previously, changes in the magnetic flux due to one circuit can effect what goes on in other circuits The changing magnetic flux induces an emf in the second circuit Mutual InductanceSuppose that we have two coils,Coil 1 with N1 turns and Coil 2 with N2 turnsCoil 1 has a current i1 which produces a magnetic flux, FB2, going through one turn of Coil 2If i1 changes, then the flux changes and an emf is induced in Coil 2 which is given byMutual InductanceThe flux through the second coil is proportional to the current in the first coilwhere M21 is called the mutual inductanceTaking the time derivative of this we getorMutual InductanceIf we were to start with the second coil having a varying current, we would end up with a similar equation with an M12We would find thatThe two mutual inductances are the same because the mutual inductance is a geometrical property of the arrangement of the two coilsTo measure the value of the mutual inductance you can use eitherorUnits of InductanceSelf InductanceSuppose that we have a coil having N turns carrying a current IThat means that there is a magnetic flux through the coilThis flux can also be written as being proportional to the currentwith L being the self inductancehaving the same units as the mutual inductanceSelf InductanceIf the current changes, then the magnetic flux through the coil will also change, giving rise to an induced emf in the coilThis induced emf will be such as to oppose the change in the current with its value given byIf the current I is increasing, thenIf the current I is decreasing, thenSelf InductanceThere are circuit elements that behave in this manner and they are called inductors and they are used to oppose any change in the current in the circuitAs to how they actually affect a circuit’s behavior will be discussed shortlyWhat Haven’t We Talked AboutThere is one topic that we have not mentioned with respect to magnetic fields Just as with the electric field, the magnetic field has energy stored in it We will derive the general relation from a special case Magnetic Field EnergyWhen a current is being established in a circuit, work has to be doneIf the current is i at a given instant and its rate of change is given by di/dt then the power being supplied by theexternal source is given byThe energy supplied is given byThe total energy stored in the inductor is thenMagnetic Field EnergyThis energy that is stored in the magnetic field is available to act as source of emf in case the current starts to decreaseWe will just present the result for the energy density of the magnetic fieldThis can then be compared to the energy density of an electric fieldR-L CircuitWe are given the following circuitand we then close S1 andleave S2 openIt will take some finite amount of time for the circuit to reach its maximum current which is given byKirchoff’s Law for potential drops still holdsR-L CircuitSuppose that at some time t the current is iThe voltage drop across the resistor is given byThe magnitude of the voltage drop across the inductor is given byThe sense of this voltage drop is that point b is at a higher potential than point c so that it adds in as a negative quantityBut what about the behavior between t = 0 and t =R-L CircuitWe take this last equation and solve for di/dtNotice that at t = 0 when I = 0 we have thatAlso that when the current is no longer changing, di/dt = 0, that the current is given byas expectedR-L CircuitWe rearrange the original equation and then integrateThe solution for this isWhich looks likeR-L CircuitAs we had with the R-C Circuit, there is a time constantassociated with R-L CircuitsInitially the power supplied by the emf goes into dissipative heating in the resistor and energy stored in the magnetic fieldAfter a long time has elapsed, the energy supplied by the emf goes strictly into dissipative heating in the resistorR-L CircuitWe now quickly open S1 and close S2The current does not immediatelygo to zeroThe inductor will try to keep the current, in the same direction, at its initial value to maintain the magnetic flux through itR-L CircuitApplying Kirchoff’s Law to the bottom loop we getRearranging this we haveand then integrating thiswhere I0 is the current at t = 0R-L CircuitThis is a decaying exponentialwhich looks likeThe energy that was stored in the inductor will be dissipated in the resistorL-C CircuitSuppose that we are now given a fully charged capacitor and an inductor that are hooked together in a circuitSince the capacitor is fully charged there is a potential difference across it given by Vc = Q / CThe capacitor will begin to discharge as soon as the switch is closedL-C CircuitWe apply Kirchoff’s Law to this circuitRemembering thatWe then have that The circuit equation then becomesL-C CircuitThis equation is the same as that for the Simple Harmonic Oscillator and the solution will be similarThe system oscillates with angular frequencyj is a phase angle determined from initial conditionsThe current is given byL-C CircuitBoth the charge on the capacitor and the current in the circuit are oscillatoryThe maximum charge and the maximum current occur p / (2w) seconds apartFor an ideal situation, this circuit will oscillate foreverL-C CircuitL-C CircuitJust as both the charge on the capacitor and the current through the inductor oscillate with time, so does the energy that is contained in the electric field of the capacitor and the magnetic field of the inductorEven though the energy content of the electric and magnetic fields are varying with time, the sum of the two at any given time is a constantL-R-C CircuitInstead of just having an L-C circuit with no resistance, what happens when there is a resistance R in the circuitAgain let us start with the capacitor fully charged with a charge Q0 on itThe switch is now closedL-R-C CircuitThe circuit now looks likeThe capacitor will start to discharge and a current will start to flowWe apply Kirchoff’s Law to this circuit and getAnd remembering thatwe getL-R-C CircuitThe solution to this second order differential equation is similar to that of the damped harmonic oscillatorThe are three different solutionsUnderdampedCritically DampedOverdampedWhich solution we have is dependent upon the relative values of R2 and 4L/CL-R-C CircuitUnderdamped:The solution to the second differential equation is thenThe system still oscillates but with decreasing amplitude, which is represented by the decaying exponentialThis solution looks likeThis decaying amplitude is often referred to as the envelopeL-R-C CircuitCritically Damped:Here the solution is given byThis solution looks likeThis is the situation when the system most quickly reachesq = 0L-R-C CircuitOverdamped:Here the solution has the formThis solution looks likewithL-R-C CircuitThe solutions that have been developed for this L-R-C circuit are only good for the initial conditions at t = 0 that q = Q0 and that i = 0

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