Review of Constants
Imagine that the parameters R, I, C, and the amplitude of the voltage V are fixed, but the frequency is very high. In this case, the current amplitude Io will also change. The frequency at which Io is at a maximum is called resonance. Find the frequency wo at which the circuit reaches resonance.
Express your answer in terms of any or all of R, I, and C.
In this problem, you will consider a series L-R-C circuit containing a resistor of resistance R, an inductor of inductance L, and a capacitor of capacitance C, all connected in series to an AC source providing an alternating voltage V=V cos(t).
Part A:
You may have solved a number of problems in which you had to find the effective resistance of a circuit containing multiple resistors. Finding the overall resistance of a circuit is often of practical interest. In this problem, we will start our analysis of this L-R-C circuit by finding its effective overall resistance, or impedance. The impedance Z is defined by Z = sqrt(R^2 + (wL - 1/(wC))^2), where R and w are the amplitudes of the voltage across the entire circuit and the current, respectively.
Part D:
What is the phase angle between the voltage and the current when resonance is reached?
Part E:
Now imagine that the parameters R, I, w, and the amplitude of the voltage V are fixed but that the value of C can be changed. This is one of the easiest parameters to change when "tuning" such (radio frequency) circuits in order to make them resonate. This is because the capacitance can be changed just by adjusting the value of a variable capacitor.
Express your answer in terms of I and w.
