RC-circuit -- Need help understanding why I get the wrong answer

[Jonas E]

Homework Statement

I have an RC circuit where C = 0.4 * 10^-6, R = 50k Ohms. The problem says: "Find the length of time required to dissipate 75% of the initially stored energy."

Homework Equations

energy 1: 1/2 * C * v(t)
energy 2) w = ∫p(t)dx, from 0 to t

The Attempt at a Solution

This is the last part of the problem. I solved everything else up to this point correctly. v(0) = 200 V, Tau = 0.02s. The initial energy stored in the capacitor is 8mJ. Now I set p(t) = v(t) / R, and integrate from 0 to t. That gives me the expression for the energy, which I then set equal to 0.25 * w(0) = 2 * 10^-3 J. However this gives me that t = 2.877ms, which is wrong according to the solutions manual. It uses the first energy formula and gets t = 13.86ms. Can someone explain why these two formulas give two different answers, or point out any mistake I have done.

 

[phyzguy]

The energy dissipated in a resistor is V^2/R, not V/R.

 

[NascentOxygen]

Your equation for energy stored in a capacitor doesn't look right.

 

[rude man]

energy 1: 1/2 * C * v(t)

This is incorrect.

 

[Jonas E]

Sorry, it should say v^2 in the energy formula, but that doesn't change anything since I used the other formula anyway. My problem is that the 1st formula (which I mistyped) gives a different answer than the second one (which is the one I used).

 

[NascentOxygen]

If you have dissipated 75% of the energy in the capacitor, then the dissipation in the resistor should be that 75% energy.

 

[rude man]

Now I set p(t) = v(t) / R, and integrate from 0 to t.

That formula is also wrong. Did you read and understand post #2?

 

[phyzguy]

JonasE:

We're trying to help, but I think we are all lost as to what you did, which formulas you used and which ones you mis-typed. Can you correct the two errors in your original post:

P = V^2 / R, not V/R
E = C*V^2/2, not C*V/2

and tell us if this fixes your issue or not? If not, what are you getting now?

 

[Jonas E]

Don't know how to edit the original post, so I'll post the edit here:

I set p(t) = V(t)^2 / R and integrate from 0 to t. This gives me the following expression: w(t) = -1/125 * (e^(-100t) - 1), which I set equal to 0.25 * 8 * 10^-3 in order to find the time. (0.25 because I need to find how long it takes to dissipate 75% of the initial energy). Solving that gives me t = ln(3/4) / (-100) = 2.877 ms. According to the solutions manual, this is wrong.

 

[phyzguy]

OK, good. What you are doing makes sense. The problem is what NascentOxygen said in Post #6. w(t) is the energy dissipated in the resistor, not the energy stored in the capacitor. So you want to set this equal to 0.75*E0, not 0.25*E0. Then both methods give the 13.86 ms. Do you see?

Also, these problems are much easier if you do the algebra with symbols, and only put in the numbers at the very end. You would then see that it doesn't matter what V(0) is, for example.

 

[Jonas E]

OK, good. What you are doing makes sense. The problem is what NascentOxygen said in Post #6. w(t) is the energy dissipated in the resistor, not the energy stored in the capacitor. So you want to set this equal to 0.75*E0, not 0.25*E0. Then both methods give the 13.86 ms. Do you see?

Also, these problems are much easier if you do the algebra with symbols, and only put in the numbers at the very end. You would then see that it doesn't matter what V(0) is, for example.

I understand now, thanks a lot for clearing it up!