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Lecture 18-1 Ways to Change Magnetic Flux B BA cos • Changing the magnitude of the field within a conducting loop (or coil). • Changing the area of the loop (or coil) that lies within the magnetic field. • Changing the relative orientation of the field and the loop. motor generator http://www.wvic.com/how-gen-works.htm Lecture 18-2How to use Faraday’s law to determine the induced current direction 1. define the direction of n ; can be any of the two normal direction, e.g. n point to right 2. determine the sign of Φ. Here Φ>0 N 3. determine the sign of ∆Φ. Here ∆Φ >0 4. determine the sign of using faraday’s law. Here <0 5. RHR determines the positive direction for EMF • If >0, current follow the direction of the curled fingers. • If <0, current goes to the opposite direction of the curled fingers. Lecture 18-3 Eddy Current A current induced in a solid conducting object, due to motion of the object in an external magnetic field. • The presence of eddy current in the object results in dissipation of electric energy that is derived from mechanical motion of the object. • The dissipation of electric energy in turn causes the loss of mechanical energy of the object, i.e., the presence of the field damps motion of the object. Lecture 18-4 Self-Inductance • As current i through coil increases, magnetic flux through itself increases. This in turn induces counter EMF in the coil itself • When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease. Self-induction B L i NB L (if flux linked) i L T m2 / A Wb / A H (henry) Faraday’s Law: dB dI L dt dt Lecture 18-5 Lecture 18-6 Solenoid: Archetypical Inductor Current i flows through a long solenoid of radius r with N turns in length l r l B 0 N i l For each turn N A r B BA 0 i r 2 l For the solenoid NB N2 2 N L 0 r 0 l r 2 i l l or 0 H / m 2 2 L 0n 2 Al Inductance, like capacitance, only depends on geometry Lecture 18-7 Potential Difference Across Inductor V I r + V I internal resistance • Analogous to a battery • An ideal inductor has r=0 - • All dissipative effects are to be included in the internal resistance (i.e., those of the iron core if any) dI 0 IR L 0 dt dI 0 IR L 0 dt Lecture 18-8 1. 2. RL Circuits – Starting Current Switch to e at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. Loop Rule: + - dI IR L 0 dt 3. Solve this differential equation I R 1 et /( L / R ) , VL L dI e t /( L / R ) dt τ=L/R is the inductive time constant T m2 / A T m2 / A s L / R V/A Lecture 18-9 Starting Current through Inductor vs Charging Capacitor q C (1 e t / RC ) IR 1 e t /( L / R ) I R e t / RC dI VL L e t /( L / R ) dt Lecture 18-10 Warm-up Which of the following statement is correct after switch S is closed ? R1 S V R2 L 1. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R1 2. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V. 3. At t = 0, the potential drop across the inductor is 0; When t = ∞, the current through R1 is V/(R1+R2) 4. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R2 Lecture 18-11 Remove Battery after Steady I already exists in RL Circuits 1. Initially steady current Io is flowing: I 0 R1 R 2. - Switch to f at t=0, causing back EMF to oppose the change. dI IR L 0 dt 3. Loop Rule: 4. Solve this differential equation I R + e t /( L / R ) dI VL L e t /( L / R ) dt like discharging a capacitor I cannot instantly become zero! Self-induction Lecture 18-12 Behavior of Inductors • Increasing Current – Initially, the inductor behaves like a battery connected in reverse. – After a long time, the inductor behaves like a conducting wire. • Decreasing Current – Initially, the inductor behaves like a reinforcement battery. – After a long time, the inductor behaves like a conducting wire. Lecture 18-13 1. 2. Energy Stored By Inductor Switch on at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. Loop Rule: IR L + dI 0 dt - 3. Multiply through by I dI I I R LI dt 2 Rate at which battery is supplying energy Rate at which energy is stored in inductor L dU m dI LI dt dt Rate at which energy is dissipated by the resistor Um 1 2 LI 2 Lecture 18-14 Where is the Energy Stored? • Energy must be stored in the magnetic field! Energy stored by a capacitor is stored in its electric field • Consider a long solenoid where 2 B 0nI , L 0n Al 2 1 2 1 1 B U m LI 0n 2 Al I 2 Al 2 2 2 0 • So energy density of the magnetic field is 2 Um 1 B um Al 2 0 1 uE 0 E 2 2 (Energy density of the electric field) area A length l Lecture 18-15 Physics 241 –Quiz A The switch in this circuit is initially open for a long time, and then closed at t = 0. What is the magnitude of the voltage across the inductor just after the switch is closed? a) zero b) V c) R / L d) V / R e) 2V Lecture 18-16 Physics 241 –Quiz B The switch in this circuit is closed at t = 0. What is the magnitude of the voltage across the resistor a long time after the switch is closed? a) zero b) V c) R / L d) V / R e) 2V Lecture 18-17 Physics 241 –Quiz C The switch in this circuit has been open for a long time. Then the switch is closed at t = 0. What is the magnitude of the current through the resistor immediately after the switch is closed? a) zero b) V / L c) R / L d) V / R e) 2V / R