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## Homework Statement

I'm trying to show that not all (half) of the energy supplied by the power supply ends up in the capacitor when the capacitor is charged through a resistor. I've looked at some of the other threads on these sorts of topics and I'd thought I'd have a go at working through it myself. The treatments of this problem that I've seen seem to consider the work done moving each charge element through a potnetial differnece but I thought I could do this another way. Anyway, it's not going well and I'd appreciate any advice.

## Homework Equations

I've included these in the working below...

## The Attempt at a Solution

It seems to me that you can start by considering the power transferred to C and to R over a given time interval and then integrate to get the energy transferred in this time.

At first, all of the p.d. is dropped across R. Let's call this V

_{0}. We can also call the initial current I

_{0}. None of the p.d. is dropped across the cap at first.

After time t, the p.d. across R is V

_{0}e

^{-t/RC}and the p.d. across the cap is V

_{0}(1-e

^{-t/RC}). The current is I

_{0}e

^{-t/RC}.

At ant time t, the power delivered to the resistor is V(t)I(t). So after some time T, the energy lost as heat in the resistor should be V

_{0}I

_{0}[tex]\int^T_0[/tex]e-

^{2t/RC}dt

Setting Vo Io as Po and with some calculating I get...

E

_{R}=(2Po/RC)(1 - e

^{-2T/RC})

It also occurred to me that the total energy delivered to the circuit would be [tex]\int^T_0[/tex]Vo Io e

^{-t/RC}dt [the total p.d. across the circuit doesn't change, only the way it is shared out changes. The current falls with time.]Evaluating this gives me

E

_{T}= (Po/RC)(1 - e

^{-T/RC})So far so good (unless I've made some stupid mistake!) Now for the energy to the capacitor. Following the same reasoning as before I get that...

E

_{C}= Po [tex]\int^T_0[/tex] e

^{-t/RC}- e

^{-2t/RC}dt

Evaluating this gives me

E

_{C}= (Po/RC)(2e

^{-2T/RC}-e

^{-T/RC}- 1)Now I'm doubtful of this result because it looks like it's got too many terms in it, but on the other hand, when I calculate E

_{C}= E

_{T}- E

_{R}I get the same result so at least if I'm wrong I'm internally consistent!

At this point my algebra fails me and I can't see how I can compare my expressions for each of the energies to show something like E

_{C}= 0.5 E

_{T}. Is this because there's something seriously wrong with my thinking?

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