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Induction: DEMONSTRATION Lecture 12 • Electrostatics *Current flows only if there is relative motion between the loop and the magnet • motion of “q” in external E-field • E-field generated by q *Current disappears when the relative motion ceases • Magnetostatics • motion of “q” and “I” in external B-field • B-field generated by “I” *Faster motion produces a greater current • Electrodynamics • time dependent B-field generates E-field – ac circuits, inductors, transformers, etc – time dependent E-field generates B-field Caution – this picture is not an example of right hand rule! 10/3/10 Induction Effects from Currents • When the switch is closed (or opened) b ⇒current induced in coil b An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing. Only components perpendicular to loop A Define the flux of the magnetic a B ⇒no current induced in coil b B field through an open surface A as: Φ B = BA cosθ Unit: 1 Weber=1Wb=1 Tm2 • Conclusion: A current is induced in a loop when: there is a change in magnetic field through it. This can happen many different ways. • How can we quantify this? 2 Faraday’s Law • Steady state current in coil a 10/3/10 ammeter 3 10/3/10 4 1 Lenz’ Law (to determine direction of induced I in loop) Faraday’s Law Restated The magnitude of the emf ε induced in a conducting loop is equal to the rate at which the magnetic flux ΦB through the loop changes. A B B An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current. Binduced always opposes the change in Opposition to Flux: the flux of B, but does not always point opposite it!!! ΔΦ B ε=− Δt B v B Direction of induced current is opposed to that from B field. 5 N S 10/3/10 v Direction of I must be to oppose change...otherwise violate conservation of energy. 6 Magnetic Flux through a Changing Area How to Change Magnetic Flux in a Coil ΔΦ B ΔB 1. B changes: =N A cosθ Δt Δt ΔΦ B ΔA 2. A changes: = NB cosθ Δt Δt Δ [ cosθ ] ΔΦ B 3. θ changes: = NBA Δt Δt ΔΦ B ΔN 4. N changes (unlikely): = BA cosθ Δt Δt 10/3/10 S Bi The minus sign indicates opposition 10/3/10 N 7 10/3/10 8 2 Energy Conservation Demo: E&M Cannon • Rate of work by applied force: P = Fv • The induced current gives rise to a net magnetic force F in the loop which opposes the motion: FL = ILB points to left • Energy is dissipated in circuit at rate P’: 9 Inductors & Inductance Symbol for inductor Induced current opposes change of in flux from change in current. This called self induced emf. Loop at right: close switch at b, current starts to flow. While dI/dt is not equal to zero, εL is induced. 10/3/10 side view 10 An induced emf, εL, appears in any coil in which the current is changing. A long solenoid can be used to produce a desired B field NΦ B L= i 1 Henry = 1H = 1 ~ 10/3/10 Self Inductance *An inductor can be used to produce a desired B field. Units of L: B F = ILB 10/3/10 *Inductance: v • Connect solenoid to a source of alternating voltage. • The flux through the area to axis of solenoid therefore changes in time • A conducting ring placed on top of the solenoid will have a current induced in it opposing this change. • There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field. • Note that it’s the off-axis component of B (the “fringe field”) that flings the ring. εL = − ΔNΦ B Δi = −L Δt Δt direction of εL opposes change with time in current. Tm 2 A X XX X X XX XX X dI/dt X XX X a b Energy Stored in an Inductor U= 1 2 LI 2 Self-Induction: Changing current through a loop induces an opposing voltage in that same loop. 11 10/3/10 12 3 Calculation of Inductance RL Circuits *Long solenoid with N turns, radius r, length L: l r N turns only depends on geometry 10/3/10 Initially, an inductor acts to oppose changes in current through it. A long time later, it acts like an ordinary connecting wire. 13 RL Circuits (ε on) Current I= ε ε 1 − e− Rt / L = 1 − e−t /τ RL R R ( ) ( 10/3/10 RL Circuits • Why does τRL increase for larger L? ) 14 a I R b ε L • Why does τRL decrease for larger R? Voltage on L VL = ε e− Rt / L = ε e−t /τ RL 10/3/10 I 15 10/3/10 16 4