Simple LR and RC circuits are familiar to generations of physics students as examples of single-exponential growth and decay in the relevant voltages, currents, and charges. An element of novelty can be introduced by connecting two (instead of one) LR coils in parallel with a battery. The resulting circuit can still be treated using little more than the basic tools (Kirchhoff's rules plus a trial exponential solution) employed in the standard LR analysis. But the solution is now a double exponential, as can be verified by constructing such a circuit.

Topics
RC circuits, Batteries, Education
1.
Boris
Korsunsky
, “
Physics challenges for teachers and students: Double closure
,”
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2.
One can dispense with R3 and trivially get a double exponential if one moves the switch into the middle branch and measures the current through the battery as a function of time. In this case, one simply obtains the sum of the standard single-exponential currents through each coil.
3.
It can be shown that B2 exceeds 4A for any positive values of L1, L2, R1, R2, and R3. This remains true if either R1 or R2 is zero.
4.
An analogous example of double exponentials is population lifetimes of species in fluorescence and radioactive decay chains, as in
L.
Moral
and
A. F.
Pacheco
, “
Algebraic approach to the radioactive decay equations
,”
Am. J. Phys.
71
,
684
–
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2003
). In the present circuit, one might imagine current cascading down from the pump ε to the two coils and exponentially seeking a new steady state whenever equilibrium is disturbed (by flipping the switch).
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5.
Another example of a double exponential with zero initial slope is the position versus time of an overdamped, undriven harmonic oscillator released from rest with a positive initial displacement.
6.
The current amplitudes in Eq. (11) have the form 0/0 for this case, but nevertheless have well-defined values. The point is that if R3 = 0, then Eq. (4) and the analogous equation for loop febc give decoupled equations for I1 and I2 with single-exponential solutions.
7.
See for example the discussion of two equal masses on two equal springs joined by a different middle spring in Sec. 11.3 of J.R. Taylor, Classical Mechanics (University Science Books, Sausalito CA, 2005).
8.
Art
Hovey
, “
Solutions to physics challenges for teachers and students: Double closure
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9.
The reason for this coincidence of the final answers for Q becomes clear if one expresses I1 in terms of dI1/dt and dI2/dt from Eq. (3) when R2 = 0, and substitutes that result into the integral in Eq. (19). One then discovers that Q only depends on the values of the currents at t = 0 and ∞, and not on their functional forms at intermediate times.
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© 2005 American Association of Physics Teachers.
2005
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