2P24- Last Time: Faraday?s Law Mutual Inductance 3P24- Faraday?s Law of Induction B d N dt ? ? =? Changing magnetic flux induces an EMF Lenz: Induction opposes change 4P24- Mutual Inductance 112 122 112 12 2 NMI N M I ? ? ? ?= 2 12 dI dt M? ?? 12 21 M MM== A current I 2 in coil 2, induces some magnetic flux ? 12 in coil 1. We define the flux in terms of a ?mutual inductance? M 12 : You need AC currents! 6P24- This Time: Self Inductance 7P24- Self Inductance What if we forget about coil 2 and ask about putting current into coil 1? There is ?self flux?: 111 11 NMILI N L I ? ?? ? ?= dI L dt ? ?? 8P24- Calculating Self Inductance N L I ? = Vs 1 H = 1 A ? Unit: Henry 1. Assume a current I is flowing in your device 2. Calculate the B field due to that I 3. Calculate the flux due to that B field 4. Calculate the self inductance (divide out I) 9P24- Group Problem: Solenoid Calculate the self-inductance L of a solenoid (n turns per meter, length A, radius R) REMEMBER 1. Assume a current I is flowing in your device 2. Calculate the B field due to that I 3. Calculate the flux due to that B field 4. Calculate the self inductance (divide out I) LN I= ? 10P24- Inductor Behavior I dI L dt ? =? L Inductor with constant current does nothing 11P24- dI L dt ? =? Back EMF I 0, 0 L dI dt ?> < I 0, 0 L dI dt ?< > 12P24- Inductors in Circuits Inductor: Circuit element which exhibits self-inductance Symbol: When traveling in direction of current: dI L dt ? =? Inductors hate change, like steady state They are the opposite of capacitors! 14P24- LR Circuit 0 i i dI VL dt IR?= =?? ? 15P24- LR Circuit 0 dI L dI LI dt R dt R IR ? ? ?? =? =?? ?? ?? ?? Solution to this equation when switch is closed at t = 0: () / () 1 t It e R ? ? ? =? :LR timeconstant L R ? = 16P24- LR Circuit t=0 + : Current is trying to change. Inductor works as hard as it needs to to stop it t=?: Current is steady. Inductor does nothing. 17P24- LR Circuit Readings on Voltmeter Inductor (a to b) Resistor (c to a) c t=0 + : Current is trying to change. Inductor works as hard as it needs to to stop it t=?: Current is steady. Inductor does nothing. 18P24- General Comment: LR/RC All Quantities Either: ( ) / Final Value( ) Value 1 t te ?? =? / 0 Value( ) Value t te ?? = ? can be obtained from differential equation (prefactor on d/dt) e.g. ? = L/R or ? = RC 19P24- Group Problem: LR Circuit 1. What direction does the current flow just after turning off the battery (at t=0+)? At t=?? 2. Write a differential equation for the circuit 3. Solve and plot I vs. t and voltmeters vs. t 21P24- Non-Conservative Fields R=100? R=10? B d d dt ? ?=? ? Es G G I=1A E is no longer a conservative field ? Potential now meaningless 22P24- This concept (& next 3 slides) are complicated. Bare with me and try not to get confused 23P24- Kirchhoff?s Modified 2nd Rule B i i d VdN dt ? ?=? ? =+ ? ? Es G G v 0 B i i d VN dt ? ??? = ? If all inductance is ?localized? in inductors then our problems go away ? we just have: 0 i i dI VL dt ? ?= ? 24P24- Ideal Inductor ? BUT, EMF generated in an inductor is not a voltage drop across the inductor! dI L dt ? =? inductor 0Vd????= ? Es G G Because resistance is 0, E must be 0! 25P24- Conclusion: Be mindful of physics Don?t think too hard doing it 30P24- Energy in Inductor 31P24- Energy Stored in Inductor dI IR L dt ? =+ + 2 dI I IR LI dt ? =+ () 22 1 2 d I IR LI dt ? =+ Battery Supplies Resistor Dissipates Inductor Stores 32P24- Energy Stored in Inductor 2 1 2 L ULI= But where is energy stored? 33P24- Example: Solenoid Ideal solenoid, length l, radius R, n turns/length, current I: 0 BnIµ= 22 o LnRlµ?= ( ) 2222 11 22 Bo ULI nRlIµ?== 2 2 2 B o B URl? µ ?? = ?? ?? Energy Density Volume 34P24- Energy Density Energy is stored in the magnetic field! 2 2 B o B u µ = : Magnetic Energy Density 2 2 o E E u ? = : Electric Energy Density 35P24- Group Problem: Coaxial Cable X I I Inner wire: r=a Outer wire: r=b 1. How much energy is stored per unit length? 2. What is inductance per unit length? HINTS: This does require an integral The EASIEST way to do (2) is to use (1) 36P24- Back to Back EMF Eric Hudson Presentation 23 Self inductance Self Inductance
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