An alternating emf source with a variable frequency $f_{\text{d}}$ is connected in series with $80.0 \mathrm{\Omega }$ resistor and $25.0 \mathrm{mH}$ inductor. The emf amplitude is $6.00 \mathrm{V}$. (a) Draw a phasor diagram for phasor $V_{\text{R}}$ (the potential across the resistor) and phasor $V_{\text{L}}$ (the potential across the inductor). (b) At what driving frequency $f_{\text{d}}$ do the two phasors have the same length? At that driving frequency, what are
(c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

Draw the approximate vector diagrams of currents in the circuits shown in the figure. The voltage applied across the points $A$ and $B$ is assumed to be sinusoidal; the parameters of each circuit are so chosen that the total current $I_0$ lags in phase behind the external voltage by an angle $\mathrm{\phi }$.

After the switch is closed in the $LC$ circuit, shown in the figure, the charge on the capacitor is sometimes zero, but at such instants, the current in the circuit is not zero. How is this behavior possible?

An LC circuit contains a $20 \mathrm{mH}$ inductor and a $50 \mathrm{\mu F}$ capacitor with an initial charge of $10 \mathrm{mC}$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$. What is the natural frequency of the circuit?

An LC circuit contains a $20 \mathrm{mH}$ inductor and a $50 \mathrm{\mu F}$ capacitor with an initial charge of $10 \mathrm{mC}$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$. At what time is the energy stored (i) completely electrical (i.e., stored in the capacitor)? (ii) completely magnetic (i.e., stored in the inductor)?