Variable capacitor

Question

I came across an interesting task in a post:

A variable capacitor, whose initial capacity is C, is charged to a voltage U and then short-circuited through a resistor R.

1. Determine the ratio of the initial energy of the capacitor and the Joule heat generated on the resistance! Let's explain the result.

2. How long and how should we change the capacity of the capacitor so that the current remains constant during the discharge of the capacitor

I appreciate any feedback you can share!

Answer 1

1) dQ/dt + (1/RC)Q = 0 so you get the standard Q=CVe^-t/RC

I is dQ/dt = -(V/R)e^-t/RC ( I into R is +)

Initial energy is 1/2 CV² and you can find Q_JH from integral of I²Rdt from 0 to infinity (same energy as stored)

2) dQ/dt =- (1/RC(t))Q such that I is constant and C(0)=C and Q(0)=CV

If I = const then Q = At+b

plug into differential equation to solve for C(t) and resolve 2 constants with the two BC's.

I get Q = (-V/R)t +CV and that C(t) = (1/R)t + C (You can only go untill charge runs out: t = RC)