Energy lost charging a capacitor - can someone check my calculations.

[MalachiK]

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 V₀. We can also call the initial current I₀. None of the p.d. is dropped across the cap at first.

After time t, the p.d. across R is V₀e^-t/RC and the p.d. across the cap is V₀(1-e^-t/RC). The current is I₀e^-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 V₀I₀\int^T_0e-^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 \int^T_0Vo 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 \int^T_0 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?

 

[willem2]

\int e^{at} dt is not equal to a e^{at}


You want the energy for time from 0 to infinity, so you have to take the limit for t->inf,
wich should be easy as all the exponentials become 0.

 

[gneill]

For the capacitor, I get

P_c = \int_0^T I_o e^{-\frac{t}{\tau}} V_o \left(1 - e^{-\frac{t}{\tau}}\right) dt

P_c = -\frac{1}{2} \tau I_o V_o \left(2 e^{-\frac{T}{\tau}} - e^{-\frac{2T}{\tau}} - 1 \right)

P_c = \frac{1}{2} \tau I_o V_o \left(1 - e^{-\frac{2T}{\tau}} + e^{-\frac{T}{\tau}} \right)

Now, τ = RC, so

P_c = \frac{1}{2} R C I_o V_o \left(1 - e^{-\frac{2T}{\tau}} + e^{-\frac{T}{\tau}} \right)

But R I_o = V_o, so

P_c = \frac{1}{2} C V_o^2 \left(1 - e^{-\frac{2T}{\tau}} + e^{-\frac{T}{\tau}} \right)

In the limit as T → ∞ this becomes

P_c = \frac{1}{2} C V_o^2

 

[MalachiK]

Doh! I am a moron.

Thanks for the help.