Last time Ampere's Law Faraday s law 1
Faraday s Law of Induction (More Quantitative) 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. ε dφ B where B S B nda N Minus sign indicates the sense of EMF: Lenz s Law Decide on which way n goes Fixes sign of B N RHR determines the positive direction for EMF
How to use Faraday s law to determine the induced current direction n n 1. define the direction of ; can be any of the two normal direction, e.g. point to right. determine the sign of Φ. Here Φ>0 3. determine the sign of Φ. Here Φ >0 N 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. 3
Today Faraday s law Inductance and RL (RLC) circuit 4
Conducting Loop in a Changing Magnetic Field Induced EMF has a direction such that it opposes the change in magnetic flux that produced it. approaching Magnetic moment created by induced currrent I repels the bar magnet. Force on ring is repulsive. moving away Magnetic moment created by induced currrent I attracts the bar magnet. Force on ring is attractive. 5
Induced Electric Field from Faraday s Law EMF is work done per unit charge: ε W / q If work is done on charge q, electric field E must be present: ε E ds nc W q Enc ds Rewrite Faraday s Law in terms of induced electric field: E nc ds dφ This form relates E and B! B B Note that E ds 0for E fields generated by charges at rest (electrostatics) since this would correspond to the potential difference between a point and itself. => Static E is conservative. The induced E by magnetic flux changes is non-conservative. 6
iclicker Question The magnetic field is decreasing, what s the direction of the induced currents in the closed rectangular loop? A. Clockwise B. Counterclockwise C. No induced currents. 7
6D-11 Jumping Ring Is there any differences in the two rings? Why one can jump up, the other can t? http://www.youtube.com/watch?v= ZL4kbBIf39s 8
iclicker Question The magnetic field is fixed, what s the direction of the induced currents in the closed rectangular loop? A. Clockwise B. Counterclockwise C. No induced currents. 9
Example At 1, 3, and 5, B is not changing. So there is no induced emf. At, B is increasing into page. So emf is induced to produce a counterclockwise current. At 4, B in decreasing into page. So current is clockwise. 10
iclicker Question A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. Both are held in place by an external force. If the current I is decreasing, what is the direction of the magnetic force on the left edge of the loop? a. Toward the right b. Toward the left c. Toward top of page d. Toward bottom of page e. No force acts on it. I 11
iclicker Question A current directed toward the top of the page and a circular loop of wire lie in the plane of the page. If a clockwise current is induced in the loop by the current I, what can you conclude about it? a. I is increasing b. I is decreasing c. I remains constant d. I is discontinuous e. Nothing can be said. I 1
Inductance emf bat R emf coil emf d mag emf 0 N R d di Increasing I increasing B emf bat emf ind R L di emf ind N r l B 0 i l L inductance, or self-inductance 0N L R d Unit of inductance L: Henry = Volt. second/ampere 13 Inductance resists changes in current
Demos: 6C-07 Energy Stored in an Inductive Circuit 14
Circuit Analysis Tips Simplify using equivalent resistors Label currents with arbitary directions If the calculated current is negative, the real direction is opposite to the one defined by you. Apply Junction Rule to all the labeled currents. Useful when having multiple loops in a circuit. Choose independent loops and define loop direction Imagine your following the loop and it s direction to walk around the circuit. Use Loop Rule for each single loop If current I direction across a resistor R is the same as the loop direction, potential drop across R is V = I R, otherwise, V = I R For a device, e.g. battery or capacitor, rely on the direction of the electric field in the device and the loop direction to determine the Potential drop across the device Solve simultaneous linear equations
Potential Difference Across Inductor + V I r internal resistance V I 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 di 0 IR L 0 16
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. +. Loop Rule: di IR L 3. Solve this differential equation 0 - di I 1 e, V L e R t /( L / R) t /( L / R) L τ=l/r is the inductive time constant L / R T m / A T m / A V / A s 17
Remove Battery after Steady I already exists in RL Circuits 1. Initially steady current I o is flowing: 3. Loop Rule: IR I R R 0 1. Switch to f at t=0, causing back EMF to oppose the change. di L 0 - + 4. Solve this differential equation t /( L / R) I e R di VL L e t /( L / R) like discharging a capacitor I cannot instantly become zero! Self-induction 18
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. 19
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. +. Loop Rule: di IR L 0-3. Multiply through by I Rate at which battery is supplying energy di I I R LI Rate at which energy is dissipated by the resistor dum Rate at which energy is stored in inductor L 1 di LI U m LI 0
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 B 0 ni, L 0 0n Al 1 1 1 B Um LI 0n Al I Al area A So energy density of the magnetic field is ue u 1 0E m Um Al 1 B 0 (Energy density of the electric field) length l 1
iclicker Question 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) V
Transformer emf loop emf N AC prim emf sec N N sec prim emf AC Energy conservation: I sec emf sec I prim emf AC I prim N N sec prim I sec 3
Two Bulbs Near a Solenoid Varying B is created by AC current in a solenoid What is the current in this circuit? mag 0 sin t emf I d emf emf cost emf R emf R 0 cos t 0 Advantage of using AC: Currents and emf s behave as sine and cosine waves. 4
Two Bulbs Near a Solenoid Add a thick wire: Loop 1: emf R I R I 0 1 1 I 1 Loop 1 Loop : R I 0 I 0 Node: I1 I I3 I 1 I1 I3 emf R 1 I Loop I 3 5
Exercise 6
Changing Area and B Simultaneously emf emf d mag mag db R BR BR dr emf due to non-coulomb electric field What is the second term due to? Magnetic force! Motional emf: F qe qv E vb emf vbl B 7
Old question: Why use HV to transport electricity? Single home current: 100 A service V wires =IR wires Transformer: emf HV I HV = emf home I home Single home current in HV: <0.1 A Power loss in wires ~ I 8
From Kirchhoff s Loop Rule Q di L 0 C (Ideal) LC Circuit I From Energy Conservation same dq Q 1 Q E LI peak const Q Q cos( t ) I dq peak. C C de 0 Q dq di LI 0 d Q 0 1 LC C Q 0 0Q peak sin( 0t ) Q di L 0 C harmonic oscillator with angular frequency 1 Natural Frequency 0 LC
LC Oscillations No Resistance = No dissipation Q 1 U E, U B LI, I C dq
Backups 31
Electric Field in a Non-uniform Ring E NC E C E NC emf E emf I R E E tot emf NC NC dl E d C const r ENC E r C d l r d 3
iclicker Question 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) V 33
iclicker Question 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) V / R 34