A Series L-R-C Circuit: The Phasor Approach (LifeSaver!!)

In this problem, you will consider a series L-R-Ccircuit, containing a resistor of resistance , an inductor of inductance ,and a capacitor of capacitance ,all connected in series to an AC source providing an alternatingvoltage .

You may have solved a number of problems in which you had tofind the effective resistance of a circuit containing multipleresistors. Finding the overall resistanceof a circuit is often ofpractical interest. In this problem, we will start our analysis ofthis L-R-C circuit by finding its effective overallresistance,or impedance. The impedance is defined by, where and are the amplitudes of the voltage across the entirecircuit and the current, respectively.

Now imagine that the parameters , , , and the amplitude of the voltage are fixedbut that the value of can be changed. This is one of the easiestparameters tochange when "tuning" such (radio frequency) circuits in order tomake them resonate. This is because the capacitance can be changedjust by moving thecapacitor plates closer or farther apart. Findthe resonance value of capacitance .Express your answer in terms of and .