Electricity and Magnetism- Concept review and Sample questions.

1P31Cllas 31: Outline Hour 1: Concept Review /Overview PRS Questions – possible exam questions Hour 2: Sample Exam Yell if you have any questions 2P31Exxa 3 Topics • Faraday’s Law • Self Inductance • Energy Stored in Inductor/Magnetic Field • Circuits • LR Circuits • Undriven (R)LC Circuits • Driven RLC Circuits • Displacement Current • Poynting Vector NO: B Materials, Transformers, Mutual Inductance, EM Waves 3P31Gennera Exam Suggestions • You should be able to complete every problem • If you are confused, ask • If it seems too hard, you aren’t thinking enough • Look for hints in other problems • If you are doing math, you’re doing too much • Read directions completely (before & after) • Write down what you know before starting • Draw pictures, define (label) variables • Make sure that unknowns drop out of solution • Don’t forget units! 4P31Maxwwell’ Equations 0 0 0 0 (Gauss's Law) (Faraday's Law) 0 (Magnetic Gauss's Law) (Ampere-Maxwell Law) ( (Lorentz force Law) in S B C S E enc C Qd dd dt d dd I dt q ε µ µε ⋅ = Φ ⋅ = − ⋅ = Φ ⋅ = + = + × ∫∫ ∫ ∫∫ ∫ EA E s BA B s F EvB) 􀁇􀁇 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 􀁇 􀁇 􀁇􀁇 􀁷 􀁶 􀁷 􀁶 5P31Gauuss’ Law: 0S inqd ε ⋅ =∫∫ EA 􀁇􀁇 􀁷 Spherical Symmetry Cylindrical Symmetry Planar Symmetry Gaussian Pillbox I 6P31­ 2 Current Sheets Ampere’s Law: .∫ =⋅ encId 0µsB 􀁇􀁇 B B X X X X X X X X X X X X X X X X X X X X X X X X X X X X B Long Circular Symmetry (Infinite) Current Sheet Solenoid Torus/Coax 7P31Faraaday’ Law of Induction ( )cos Bdd N dt dN BA dt θ ε Φ = ⋅ =− =− ∫ E s 􀁇 􀁇 􀁶 Induced EMF is in direction that opposes the change in flux that caused it Lenz’s Law: Ramp B Rotate area in field Moving bar, entering field 8P31PPR Questions: Faraday’s & Lenz’s Law Class 21 9P31Seel Inductance & Inductors dIL dtε= − I L NL I Φ = When traveling in direction of current: Notice: This is called “Back EMF” It is just Faraday’s Law! 10P31-Energy Stored in Inductor 21 2LUL I = : Magnetic Energy Density Energy is stored in the magnetic field: 2 2B o B u µ = 11P31-LR Circuit t=0+: Current is trying to change. Inductor works as hard as it needs to in order to stop it t=∞: Current is steady. Inductor does nothing. Readings on VoltmeterInductor (a to b)Resistor (c to a) c 0dIL dtIRε =− − 12P31-General Comment: LR/RC All Quantities Either: ( )/Final Value( ) Value 1 tt e τ−= − /0Value( ) Value tt e τ−= τ can be obtained from differential equation (prefactor on d/dt) e.g. τ = L/R or τ = RC 13P31-PRS Questions: Inductors & LR Circuits Classes 23, 25 14P31-Undriven LC Circuit Oscillations: From charge on capacitor (Spring) to current in inductor (Mass) 0 1 LC ω = 15P31-Damped LC Oscillations Resistor dissipates energy and system rings down over time LQ n R ωπ== 16P31-PRS Questions: Undriven RLC Circuits Class 25 AC Circuits: SummaryElement V vs I0 Current vs. Voltage Resistance-Reactance (Impedance) Resistor 0 0V R = I R In Phase R = R Capacitor 0 0C IV ωC = Leads (90º) 1XC ωC = Inductor 0 0LV I ωL= Lags (90º) X L =ωL L32 -1718P31-Driven RLC Series Circuit Now Solve: S R L CV V V V= + + I(t) 0I 0RV0LV0CV0SVNow we just need to read the phasor diagram! VS 19P31-Driven RLC Series Circuit 0I 0RV 0LV 0CV 0SV 2 2 2 2 0 0 0 0 0 0( ) ( )S R L C L CV V V V I R X X IZ = + − = + − ≡ ϕ 1tan L CX X Rφ − −⎛ ⎞ = ⎜ ⎟⎝ ⎠ 2 2( )L CZ R X X= + −0 0 SVI Z = ( )0 sinS SV V tω= 0() sin( )It I tωϕ= − Impedance 20P31-Plot I, V’s vs. Time 0 1 2 3 0 +φ -π/2 +π/2 VS Time (Periods) 0VC 0VL 0VR 0I( ) ( ) ( ) ( ) ( ) 0 0 0 2 0 2 0 () sin () sin () sin () sin () sin R L L C C S S It I t Vt IR t Vt IX t Vt IX t Vt V t π π ω ω ω ω ω ϕ = = = + = − = + 1tan L CX X Rφ − −⎛ ⎞ = ⎜ ⎟⎝ ⎠ 21P31-Resonance 0 0 0 2 2 1; , ( ) L C L C V VI X L XZ CR X X ω ω = = = = + − C-like: φ < 0 I leads 0 1 LCω = L-like: φ > 0 I lags On resonance: I0 is max; XL=XC; Z=R; φ=0; Power to R is max 22P31-Average Power: Resistor () 2 2 2 0 2 2 0 2 1 0 2 () sin ( ) sin ( ) P I tR I t R IR t IR ωϕ ωϕ <>=< > =< − > = < − > = 23P31-PRS Questions: Driven RLC Circuits Class 26 24P31-Displacement Current 0 E d ddQ Idt dtε Φ = ≡ 0 0 0 E QE Q EA A ε εε = ⇒ = = Φ 0 0 0 0 ( )encl d C E encl d I I dI dt µ µ µε ⋅ = + Φ = + ∫ Bs 􀁇 􀁇 􀁶 Capacitors, EM Waves 25P31-Energy Flow 0µ × = EBS 􀁇 􀁇􀁇 Poynting vector: • (Dis)charging C, L • Resistor (always in) • EM Radiation 26P31-PRS Questions: Displacement/Poynting Class 28 27P31-SAMPLE EXAM: The real exam has 8 concept, 3 analytical questions 28P31-Problem 1: RLC Circuit 1. Write a differential equation for the current in this circuit. 2. What angular frequency ωres would produce a maximum current? 3. What is the voltage across the capacitor when the circuit is driven at this frequency? Consider a circuit consisting of an AC voltage source: V(t)=V0sin(ωt) connected in series to a capacitor C and a coil, which has resistance R and inductance L0. 29P31-Solution 1: RLC Circuit 2. Maximum current on resonance: ( ) 2 2 0 0 cos S S dI QV IR L dt C dI d I I dR L Vdt dt C dt V tω ω − − − = + + = =( )0 sinSVV tω= 1. Differential Eqn: 0 1 res L Cω = 30P31-Solution 1: RLC Circuit 0 0C CV I X = ( )0 sinSV V tω= 3. Voltage on Capacitor What is I0, XC? 0 0 0 (resonance) V VI Z R = = 0 01 C LC LX C C Cω = = = 0 0 0 0C C V LV I X R C = = ( )( ) ( ) 0 0 2 cos sin C C VV t V t π ω ω = − = − 31P31-Problem 1, Part 2: RLC Circuit 4. Did the inductance increase or decrease? 5. Is the new resonance frequency larger, smaller or the same as before? 6. Now drive the new circuit with the original ωres. Does the current peak before, after, or at the same time as the supply voltage? Continue considering that LRC circuit. Insert an iron bar into the coil. Its inductance changes by a factor of 5 to L=Lcore 32P31-4. Putting in an iron core INCREASES the inductance 5. The new resonance frequency is smaller 6. If we drive at the original resonance frequency then we are now driving ABOVE the resonance frequency. That means we are inductor like, which means that the current lags the voltage. Solution 1, Part 2: RLC Circuit 33P31-Problem 2: Self-Inductance The above inductor consists of two solenoids (radius b, n turns/meter, and radius a, 3n turns/meter) attached together such that the current pictured goes counter-clockwise in both of them according to the observer. What is the self inductance of the above inductor? 􀁁 34P31-Solution 2: Self-Inductance b X X X X X X X X X X X X X X X X X X X X X X X X X X a 􀁁 n 3n Inside Inner Solenoid: ( )0 0 3 4 d Bl nlI nlI B nI µ µ ⋅ = = + ⇒ = ∫ Bs 􀁇 􀁇 􀁶 Between Solenoids: 0B nI µ= ( ) ( ) ( ) 2 2 2 0 02 2 2 Volume 2 4 2 2 o o o BU nI nI a b a µ µ µπ π µ µ = = + − i 􀁁 􀁁 35P31-Solution 2: Self-Inductance b X X X X X X X X X X X X X X X X X X X X X X X X X X a 􀁁 n 3n ( ) { } 2 0 2 215 2 o nIU a bµ π µ = +􀁁 21 2U L I = ( ) { } 2 0 2 215 o nL a bµ π µ ⇒ = +􀁁 Could also have used: NL I Φ = 36P31-Problem 3: Pie Wedge 1. If the angle θ decreases in time (the bar is falling), what is the direction of current? 2. If θ = θ(t), what is the rate of change of magnetic flux through the pie-shaped circuit? Consider the following pie shaped circuit. The arm is free to pivot about the center, P, and has mass m and resistance R. 37P31-Solution 3: Pie Wedge 2) θ = θ(t), rate of change of magnetic flux? 1) Direction of I? Lenz’s Law says: try to oppose decreasing flux I Counter-Clockwise (B out) 2 2 2 2 aA a θ θπ π ⎛ ⎞ = =⎜ ⎟⎝ ⎠ ( ) 2 2 Bd d d aBA Bdt dt dt θΦ = = 2 2 Ba d dt θ = 38P31-Problem 3, Part 2: Pie Wedge 3. What is the magnetic force on the bar (magnitude and direction – indicated on figure) 4. What torque does this create about P? (HINT: Assume force acts at bar center) 39P31-Solution 3, Part 2: Pie Wedge 4) Torque? 3) Magnetic Force? 2 aFτ=× ⇒ =τ rF 􀁇􀁇􀁇 d Id= ×F sB 􀁇 􀁇􀁇 24 4 Ba d Rdt θ = F IaB = (Dir. as pictured) 23 2 Ba d F Rdt θ = 1 BdI R R dt ε Φ = = 21 2 Ba d R dt θ = (out of page) 40P31-Problem 4: RLC Circuit 1. What energy is currently stored in the magnetic field of the inductor? 2. At time t = 0, the switch S is thrown to position b. By applying Faraday's Law to the bottom loop of the above circuit, obtain a differential equation for the behavior of charge Q on the capacitor with time. The switch has been in position a for a long time. The capacitor is uncharged. 41P31-Solution 4: RLC Circuit 1. Energy Stored in Inductor 2 21 1 2 2 U L I L R ε⎛⎞ = = ⎜⎟⎝⎠ 2. Write Differential Equation I 0dI QL dt C− − = 2 2 0dQ d Q QI Ldt dt C = ⇒ + = +Q 42P31-Problem 4, Part 2: RLC Circuit 3. Write down an explicit solution for Q(t) that satisfies your differential equation above and the initial conditions of this problem. 4. How long after t = 0 does it take for the electrical energy stored in the capacitor to reach its first maximum, in terms of the quantities given? At that time, what is the energy stored in the inductor? In the capacitor? 43P31-Solution 4: RLC Circuit 3. Solution for Q(t): ()max () sin Qt Q tω= 4. Time to charge capacitor 1 LC ω= max 0Q I R εω = = Charge 2 2 4 2 T LC T LC Tπ ππω = = ⇒ = = Energy in inductor = 0 Energy in capacitor = Initial Energy: 21 2 U L R ε⎛⎞ = ⎜⎟⎝⎠ ⇒max LCQ R ε = 44P31-Problem 5: Cut Circuit 1. After a time t = t0, a charge Q = Q0 accumulates at the top of the break and Q= -Q0 at the bottom. What is the electric field inside the break? 2. What is the magnetic field, B, inside the break as a function of radius r