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Inductance

Inductance. General Physics II, Spring 2011 Brian Meadows University of Cincinnati. Outline. What is Inductance and what are the units? Self Inductance of solenoid Series Inductance and Resistance (L-R) circuits Energy in magnetic field Mutual Inductance. Outline.

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Inductance

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  1. Inductance General Physics II, Spring 2011 Brian Meadows University of Cincinnati Brian Meadows, U. Cincinnati.

  2. Outline • What is Inductance and what are the units? • Self Inductance of solenoid • Series Inductance and Resistance (L-R) circuits • Energy in magnetic field • Mutual Inductance Brian Meadows, U. Cincinnati

  3. Outline • What is Inductance and what are the units? • Self Inductance of solenoid • Series Inductance and Resistance (L-R) circuits • Energy in magnetic field • Mutual Inductance Brian Meadows, U. Cincinnati

  4. Inductance • When a current flows in a circuit, it produces a magnetic field linkage • E.g. a current in a solenoid of N turns produces a magnetic field B that creates a flux NB in itself. • A flux linkage with another circuit can also be established • The former is “self inductance”, the latter is “mutual inductance” (n = N/L) i B = 0ni NB = nLBA = 0n2LAi L Brian Meadows, U. Cincinnati

  5. Inductance and Self Inductance • The flux linkage NBis proportional to the current i that produces it. NB = L i (analogous to C in q= CV ) • The unit of inductance is the henry (H) [L] = [B]/[A] so 1 H = 1 T.m2/A • The self-inductance of a solenoid of length Land area A with n turns per unit length an is given by L = 0 n2 LA Brian Meadows, U. Cincinnati

  6. Inductance • When a current in a circuit changes, it produces a changing magnetic field dB/dt = 0 • If the magnetic field changes, then a “back EMF” e is induced in a direction that will always oppose the change in current that is causing the field to change. • If i comes in as illustrated, and di/dt>0, then e is in direction indicated. i - + e = L di/dt Brian Meadows, U. Cincinnati

  7. Outline • What is Inductance and what are the units? • Self Inductance of solenoid • Series Inductance and Resistance (L-R) circuits • Energy in magnetic field • Mutual Inductance Brian Meadows, U. Cincinnati

  8. Series R-C Circuit (reminder) • Apply Kirchoff rule 2 to series circuit: VC= Q/C • Start at bottom right and go clockwise: • Solve this equation: • Charge rises slowly to peak value - + -q(t) +q(t) i=dq/dt + C R i=dq/dt VR= iR -  Brian Meadows, U. Cincinnati

  9. Series L-R Circuit • Apply Kirchoff rule 2 to series circuit: VL= L di/dt • Start at bottom right and go clockwise: • Solve this equation: • Current rises slowly to peak value - + i + i L R VR= iR  - Open switch (no effect) Brian Meadows, U. Cincinnati

  10. Current build-up in LR circuit This is reduced by factor “e” each time interval L/R i(t) i(1)=e/R i(0)=0 t Circuit “time constant” is L/R. This is zero if L=0. Brian Meadows, U. Cincinnati

  11. Now remove the EMF and allow current to drop: VL= L di/dt • Start at bottom right and go clockwise: • Solve this equation: • Current drops slowly from initial value to - + i + i L R VR= iR - Close switch (shorts EMF so   0) Brian Meadows, U. Cincinnati

  12. Current decay in LR circuit i(t) i(0)=e/R This is reduced by factor “e” each time interval L/R i(1)=0 t Circuit “time constant” is L/R. This is zero if L=0. Brian Meadows, U. Cincinnati

  13. Outline • What is Inductance and what are the units? • Self Inductance of solenoid • Series Inductance and Resistance (L-R) circuits • Energy in magnetic field • Mutual Inductance Brian Meadows, U. Cincinnati

  14. Energy in Magnetic Field • Energy is required to build up current (and B field) in L: VL= L di/dt • Power P = i VLso energy is • This leads to: • Recall, for a capacitor, electric field energy is - + i + i L R VR= iR  - Open switch (no effect) Brian Meadows, U. Cincinnati

  15. Outline • What is Inductance and what are the units? • Self Inductance of solenoid • Series Inductance and Resistance (L-R) circuits • Energy in magnetic field • Mutual Inductance Brian Meadows, U. Cincinnati

  16. Mutual Inductance – Pair of Coils • Flux linkage due to coil 1 in coil 2, N21 2/ i1 so • Likewise • Therefore Brian Meadows, U. Cincinnati

  17. Example – Solenoid and Short Coil • Flux linkage due to solenoid (s) in coil (c) is Nc s c = M is • This flux linkage is • Therefore Turns/m = n Field B = 0 n is Area A=r12 Nc total turns Brian Meadows, U. Cincinnati

  18. Example • Solenoid has n=2000 turns/m and r=5 cm, coil has Nc=50 turns. What is e in coil if dis/dt = 0.1 A/s ? • So • Therefore, EMF induced in coil is Turns/m = n Field B = 0 n is Area A=r12 Nc total turns Brian Meadows, U. Cincinnati

  19. Example - Transformer • Has “primary (NP turns) and “secondary” (NS turns) coils. • Iron form ensures that virtually ALL flux through primary also passes through secondary. • fP/NPandfS/NS • dfP /dtNPanddfS /dt/NS • So eS/eP= NS / NP Brian Meadows, U. Cincinnati

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