Energy stored on a capacitor

[jesuslovesu]

Homework Statement

Express the energy stored on a capacitor in terms of L, R, V in a standard LRC circuit. The circuit is at resonance.

Homework Equations

E = 1/2 CV^2

The Attempt at a Solution

Well I start with substituting C with something else.
1) C = 1/(Lw^2)
So E = 1/2 V^2/(Lw^2)

2) That's where I begin to have problems. I've then tried w = 2pi*f and f = R/L

Therefore E = V^2 * L /(2R^2 * 4pi^2). Unfortunately, I'm close to the answer but the 4pi^2 term is where my problem lies. Anyone know where I went wrong?

 

[jesuslovesu]

Cancel that, I figured it out

 

[m2003]

Your approach is correct, but you have made a mistake in your calculation. Let's go through the steps again:

1. We know that at resonance, the inductive reactance (XL) and the capacitive reactance (XC) are equal:
XL = XC
2. We also know that XL = wL and XC = 1/wC
3. Substituting these values into the equation, we get:
wL = 1/wC
4. Solving for w, we get:
w = 1/sqrt(LC)
5. Now, we can substitute this value of w into the equation for energy stored on a capacitor:
E = 1/2 CV^2
E = 1/2 (1/wC) V^2
E = 1/2 (1/(1/sqrt(LC))) V^2
E = 1/2 (sqrt(LC)) V^2
6. Finally, we can use the substitution f = 1/(2pi*sqrt(LC)) to get the final equation:
E = 1/2 (4pi^2f^2L) V^2

So, the correct answer is E = (2pi^2f^2L)V^2. The 4pi^2 term should not be there.

I hope this helps!