Question Two metal rings lie side-by-side on a table. Current in the left ring runs clockwise and is increasing with time. This induces a current in the right ring. This current runs
A) Clockwise B) Counterclockwise
when viewed from above Two Bulbs Near a Solenoid
Add a thick wire: I1
Loop 1 Loop 1: emf R1I1 R2I2 0
Loop 2: R2I2 0 I2 0 I2 Node: I1 I2 I3 I1 I3 emf I1 Loop 2 R1 I3 Exercise Faraday’s Law and Motional EMF
‘Magnetic force’ approach: Ftot qE qv B
E vB emf vB L I Use Faraday law: d emf mag dt
mag B A B Lv t
emf lim mag vB L I t 0 t Faraday’s Law and Generator
d emf mag dt B nˆA Bwhcos
t Bwhcos t I d d emf mag Bwh cos t dt dt emf Bwh sin t Inductance d emf mag R dt emfbat emfcoil
N 2 dI emf 0 R2 d dt Increasing I increasing B dI emf L ind dt L – inductance, or self-inductance N 2 L 0 R2 d R Unit of inductance L: emfbat Henry = Volt.second/Ampere emfind Inductance resists changes in current Current in RL Circuit
Vbattery Vresistor Vinductor 0 dI emf RI L 0 battery dt I(t) a bect
ct ct emfbattery Ra Rbe Lbce 0 emf R a battery Rb Lbc c R L R emf t I(t) battery be L R emf If t is very long: I(t ) battery R Current in RL Circuit
R emf t I(t) battery be L R If t is zero: I(0) 0 emf I(0) battery b 1 0 R emf b battery R Current in RL circuit: R emf t I(t) battery 1 e L R Time Constant of an RL Circuit
Current in RL circuit: R emf t I(t) battery 1 e L R
Time constant: time in which exponential factor drops e times
R t 1 L L R Current in an LC Circuit
Vcapacitor Vinductor 0
Q dI dQ L 0 I C dt dt d 2Q Q LC 0 dt 2 Q a bcos ct
a bcos ct LC bc2 cos ct 0 a=0 1 c LC t t Q bcos Q Q cos LC 0 LC Current in an LC Circuit
t Q Q cos 0 LC dQ I dt
Current in an LC circuit Q t I 0 sin LC LC
Period: T 2 LC Frequency: f 1/ 2 LC Energy in an LC Circuit Q 2 Initial energy stored in a capacitor: 2C 2 Q0 At time t=0: Q=Q0 U cap 2C 1 At time t= LC : Q=0 U LI 2 2 sol 2 1/4 of a period
System oscillates: energy is passed back and forth between electric and magnetic fields. Energy in an LC Circuit
What is maximum current?
At time t=0: Q2 U U U 0 total el mag 2C At time t= LC : 2 1 LI 2 2 max
2 1 2 Q0 Q0 LImax I 2 2C max LC Energy in LC Circuit
U Uelectric Umagnetic (No dissipation in this circuit)
2 1 Q 1 2 dU d( ) d( LI ) Q dQ dI 2 C 2 LI 0 dt dt dt C dt dt dQ I As capacitor loses charge, current increases dt As capacitor gains charge, current increases
Q dI L 0 C dt Same equation as obtained via considering potential differences Non-ideal LC Circuit
Vcapacitor Vinductor VR 0
Q dI RI L 0 C dt
Is charge being lost? Changing Area and B Simultaneously
2 mag B R d emf mag dt dB dR emf R2 B2 R dt dt
emf due to non-coulomb electric field Motional emf: What is the second term due to? F qE qv B Magnetic force! E vB emf vBL Faraday’s Law: Applications
Stealing power