Solving LCR Circuit: Find C, I_rms, I_rms at Resonance

[tomrja]

Homework Statement

Consider a series LCR circuit with R= 69.0 Ω and L= 0.100 H, driven by a sinusoidal emf with E_rms= 6.70 V at frequency f= 250 Hz. The sinusoidal current leads the emf by 54.0 degrees.

a) Calculate the capacitance C.
b) What is the rms current in the circuit?
c) If the frequency of the emf is changed to the resonant frequency of the circuit, what is the rms current?

Homework Equations

tan(phi)=(WL-(1/WC))/R

I_p= V^p/Z = V_p/sqrt(R^2+(X_L-X_C)^2)

I_rms=I_P/sqrt(2)

W=2*pi*f

W_o=1/sqrt(LC) resonant frequency

The Attempt at a Solution

I solved tan(phi)=(WL-(1/WC))/R for C and got C=1/(W(WL-Rtan(phi)) then plugged in all given info to solve for C. It says that the answer is wrong and I am assuming that I plugged in the wrong phi. "The sinusoidal current leads the emf by 54.0 degrees." I don't know what this means. I have not started on the other two parts of the problem because I need to find C first. What am I doing wrong? Thanks!

 

[cupid.callin]

Hi tomrja
(have a phi: φ)

tan(phi)=(WL-(1/WC))/R

i didn't read all of your post but the thing i quoted is wrong

$$tan\phi = \frac{\frac{1}{wC} - wL}{R}$$

thats because current in capacitor leads voltage by 90 and lags in inductor by 90

 

[cupid.callin]

and for i_RMS use

i_RMS = E_RMS/Z

 

[tiny-tim]

hi tomrja!

"The sinusoidal current leads the emf by 54.0 degrees." I don't know what this means.

it means that if V = V_maxsinωt, then I = I_maxsin(ωt + 54°)

in other words, the impedance is Z = |Z|e^i54π/180

 

[rwisz]

It appears that you have correctly set up the equation for calculating the capacitance C. However, the given information of the current leading the emf by 54.0 degrees means that the phase angle phi is equal to -54.0 degrees. This is because the current in an LCR circuit with a pure inductor (L) and capacitor (C) will lag behind the emf by 90 degrees, and since the given current is leading the emf, it will have a negative phase angle. Therefore, to solve for C, you would need to use tan(phi) = -(WL - (1/WC))/R. Once you have the correct value for C, you can proceed to solve for the rms current in the circuit and the rms current at resonance using the appropriate equations. Remember to convert the frequency f to angular frequency W using W = 2*pi*f. I hope this helps. Good luck with your calculations.