# 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 (from the 3rd reference link: “charge is given by “, when we treat and as a series of capacitors, the sign of is reversed against the plate sign).

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

**(2)**

**(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) & (6)**

I don’t know.