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NOTES Faraday’s Law of Induction The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time. dΦ B ∫ Enc i ds = − dt where N Minus sign indicates the sense of EMF: Lenz’s Law • Decide on which way n goes Fixes sign of ϕB • RHR determines the positive direction for EMF N Ways to Change Magnetic Flux • 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 Other Examples of Induction + - Switch has been open for some time: Switch is just closed: Nothing happening EMF induced in Coil 2 + - Switch is just opened: EMF is induced again Switch is just closed: EMF is induced in coil - + Back emf (counter emf) MUTUAL INDUCTANCE BETWEEN TWO COILS ©2008 by W.H. Freeman and Company Eddy Currents 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. 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 (if flux linked) (henry) Faraday’s Law: NOTES Inductance of a Solenoid: Basic Inductor Geometry Current i flows through a long solenoid of radius r with N turns in length l For each turn For the solenoid or Inductance, like capacitance, only depends on geometry (if made of conductor and air) Potential Difference Across Inductor +V ΔV internal resistance I • “Analogous” to a battery • An ideal inductor has r =0 V=0 • All dissipative effects are to be included in the internal resistance (i.e., those of the iron core if any) Energy Stored By Inductor 1. Switch on at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + 2. Loop Rule: - 3. Multiply through by I Rate at which energy is stored in inductor L Rate at which battery is supplying energy Rate at which energy is dissipated by the resistor NOTES 6C07 ENERGY STORED IN AN INDUCTOR 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 area A • So energy density of the magnetic field is length l (Energy density of the electric field) RL Circuits – Starting Current 1. Switch to e at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. 2. Loop Rule: 3. Solve this differential equation τ=L/R is the inductive time constant + - GROWTH AND DECAY OF CURRENT OF AN RL CIRCUIT 6C-05 Starting and Decay Currents through an Inductor and a Capacitor Remove Battery after Steady I already exists in RL Circuits 1. Initially steady current Io is flowing: 2. Switch from e to f at t=0, causing back EMF to oppose the change. + 3. Loop Rule: 4. Solve this differential equation I cannot instantly become zero! Self-induction like discharging a capacitor Behavior of Current through Inductors as Function of Time • 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. Superconductor Repels External B field when T<TC D E M O MAGNET LEVITATION ABOVE A SUPERCONDUCTOR NOTES Physics 241 Extra Quiz 3 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 Physics 241 Extra Quiz 3 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 Warm up Quiz for AC Which of the following statement is true? A. B. C. D.