# Todai Entrance Exam: Subject 2013 – Problem 1

There are 2 ways to attack this problem. For the first way, we attack the differential equation subject to .

For the second way, we combine the series of capacitors and into one capacitor , and solve this circuit with the charge conservation equation .

As this is a circuit, as , also approaches . Remember the charge conservation equation:

**(2) **Reference: 4 & 5. Here the combined capacitor is an active component (voltage source), so will follow the sign of the current through R for convenience.

**(3)**

When (see slide 8):

The total potential energy stored in and at is:

**(4)**

Reference: A negative charge flows from the negative plate of to the negative plate of , which forces the positive plate of to be more positive by having another negative charge flows from the positive plate of to the positive plate of , which makes the positive plate of to be less positive. So the voltage of keeps decreasing while the voltage of keeps increasing, the voltage difference between the twos keeps decreasing, hence the decrease in energy (which is proportional to the voltage difference).

**(5) **

Thanks to Wikipedia, the equivalent circuit without non-linear elements for is a battery with nonzero internal resistance , which turns the circuit into RC circuit. Here:

**(6)**