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*From*: Carl Mungan <mungan@usna.edu>*Date*: Wed, 8 Apr 2020 09:13:00 -0400

This problem is for the intro E&M course for majors. I don’t remember having seen this problem before so I am particularly interested in the pedagogy of it - i.e. what do you think about it and what can be learned from it? I feel like there’s more to be said about it.

PROBLEM: (All components are to be assumed ideal. If you think it would be interesting to relax this assumption, go for it.)

Find the mutual inductance M of a transformer. Label everything on the input side of the transformer by a subscript “1” and everything on the output side by “2”. There are a few components connected in a single series loop on each side. On the input side there is a 100-V dc battery, a 20-ohm resistor, and a switch. On the output side there is a 200-ohm resistor and a current integrator. In our case, the switch in the primary has been open for a long time. We close it at t=0 and start the current integrator. We keep measuring for a long time. The resulting charge Q is 325 uC.

SOLUTION:

The voltage rise across the mutual inductance equals the voltage drop across the resistor on the output side so M dI1/dt = I2 R2 = R2 dQ/dt => M dI1 = R2 dQ. Integrate both sides from t=0 to t=infinity to get M (100 V)/(20 ohm) = (200 ohm) (325 uC) = > M = 13 mH.

What about the self inductance L2 of the output coil? It plays no role. The voltage across it is L2 dI2/dt which integrates to zero because I2 is zero both initially and finally.

DISCUSSION: I have two issues so far.

1. What is the sign convention for M dI/dt using the Voltage Loop Rule? I know the conventions for batteries, capacitors, resistors, and self-inductances but what is the convention for mutual inductances? Assume I tell you the direction of guessed current in every branch and the direction I choose to go around a loop.

2. I usually try to avoid the term “voltage drop” (or “rise”) and instead prefer “potential drop.” But is it preferable to use “voltage” when referring to inductors since there is no potential inside an inductor? (Or many books use “emf” when referring to inductors.) But “drop” or “rise” is really referring here to the sign being negative or positive. How do you handle this issue in the intro course?

-----

Carl E. Mungan, Professor of Physics 410-293-6680 (O) -3729 (F)

Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363

mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/

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