Self-Inductance RL Circuits Energy in a Magnetic Field Mutual Inductance Oscillations in an LC Circuit When the switch is closed, the battery (source emf) starts pushing electrons around the circuit. The current starts to rise, creating an increasing magnetic flux through the circuit. This increasing flux creates an induced emf in the circuit. The induced emf will create a flux to oppose the increasing flux. The direction of this induced emf will be opposite the source emf. This results in a gradual increase in the current rather than an instantaneous one. The induced emf is called the self-induced emf or back emf. When I changes, an emf is induced in the coil. If I is increasing (and therefore increasing the flux through the coil), then the induced emf will set up a magnetic field to oppose the increase in the magnetic flux in the direction shown. If I is decreasing, then the induced emf will set up a magnetic field to oppose the decrease in the magnetic flux. d E N B L dt dI B E B L dt B I d dI E N B L L dt dt N L B (Henry=H=V.s/A) I E L L dI dt A N B nI I 0 0 l N l NA BA I B 0 l N N 2 A L B L n2V I 0 l 0 N nl V Al Inductors are circuit elements with large self- induction. A circuit with an inductor will generate some back- emf in response to a changing current. This back-emf will act to keep the current the way it used to be. Such a circuit will act “sluggish” in its response. When S2 is at a dI E IR L 0 dt E L I 1 et / R R When S2 is at b dI IR L 0 dt dI E L L dt t / I Iie When the switch is closed, the current through the circuit exponentially approaches a value I = E / R. If we repeat this experiment with an inductor having twice the number of turns per unit length, the time it takes for the current to reach a value of I / 2 1. increases. 2. decreases. 3. is the same. dI The energy stored in the inductor E IR L 0 dt dU dI LI dI dt dt IE I 2R LI dt I I U dU LIdI L IdI 0 0 Battery Inductor 1 2 Power Power U LI Resistor 2 Power 2 2 2 1 2 1 2 B B L 0n Al U LI n Al Al For energy density, 2 2 0 n 2 consider a solenoid 0 0 B nI 0 U B2 uB Al 20 BdA B dA ldr b I Il b dr Il b BdA 0 ldr 0 0 ln B a 2r 2 a r 2 a 1 Ul bLI 2 L B 0 ln2 I 2 a lI 2 b U 0 ln 4 a A change in the current of one circuit can induce an emf in a nearby circuit N212 M12 I1 d d M I dI 12 12 1 1 E2 N2 N2 M12 dt dt N2 dt dI E M 2 1 21 dt It can be shown that: M12 M 21 M dI2 dI1 E1 M E M dt 2 dt N B B I 0 l N N BA N N A M H BH H H B I I 0 l Provide a transfer of energy between the capacitor and the inductor. Analogous to a spring- block system Assume the capacitor is fully charged and thus has some stored energy. When the switch is closed, the charges on the capacitor leave the plates and move in the circuit, setting up a current. This current starts to discharge the capacitor and reduce its stored energy. At the same time, the current increases the stored energy in the magnetic field of the inductor. When the capacitor is fully discharged, it stores no energy, but the current reaches a maximum and all the energy is stored in the inductor. The current continues to flow and now starts to charge the capacitor again, this time with the opposite polarity. When the capacitor becomes fully charged (with the opposite polarity), the energy is completely stored in the capacitor again. The e-field sets up the current flow in the opposite direction. Then the cycle completes itself in reverse. At some arbitrary time, t: Q2 1 U U U LI 2 C L 2C 2 dU 0 dt Q d 2Q L 0 C dt 2 Q Qmax cost I Qmax sint Imax sint 1 Natural frequency of oscillation LC 2 2 Q LI max U U U max cos2 t sin 2 t C L 2C 2 2 2 Q LI max max 2C 2 Q 2 U max cos2 t sin 2 t 2C Q 2 U max 2C 1 1 f 106 Hz a) f = ? 2 2 LC 2 2.81103 91012 b) Qmax = ? and Imax = ? 12 10 Qmax CE 910 12 1.0810 C 4 Imax Qmax 2fQmax 6.7910 A c) Q(t) = ? and I(t) = ? 10 6 Q Qmax cost 1.0810 Ccos2 10 t 4 6 I Imax sint 6.7910 Asin2 10 t Q 2 d) U = ? U max 6.481010 J 2C RLC Circuit - A more realistic circuit The resistor represents the losses in the system. Can be oscillatory, but the amplitude decreases. Rt 2L Q Qmax e cosd t 2 1 2 1 R d LC 2L R 4L C Reading Assignment . Chapter 33 - Alternating Current Circuits WebAssign: Assignment 10 .