Exercise

Can you have an $AC$ series circuit in which there is a phase difference of (a) $180°$, (b) $120°$ between the emf and the current? 

Explain the meaning of ‘reactance’. Derive expression for the reactance of an inductor $\mathrm{L}$ connected across an $\mathrm{AC}$ source. Give its unit. Show that no power is dissipated in an inductor when $\mathrm{AC}$ passes through it.

A coil of inductance $5.0 \mathrm{mH}$ and negligible resistance is connected to the oscillator of the previous problem. Find the peak currents in the circuit for $\omega =100 {\mathrm{s}}^{-1}$, $500 {\mathrm{s}}^{-1}$, $1000 {\mathrm{s}}^{-1}$.

An ac generator has emf $\mathcal{E}={\mathcal{E}}_m \sin \left({\omega }_{d}t-\pi /4\right),$ where
$\mathcal{E}=25.0 \mathrm{V}$ and  ${\omega }_d=270 \mathrm{rad}/\mathrm{s}.$ The current produced in a connected circuit is $i\left(t\right)=I \sin \left({\omega }_{d}t-3\pi /4\right),$ where $I=620 \mathrm{mA}$ . At what time after $t=0$ does
(a) the generator emf first reach a maximum and (b) the current first reach a maximum?
(c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer.
(d) What is the value of the capacitance, inductance, or resistance, as the case may be?

Obtain an expression for Impedance and current in series LCR circuit. Deduce an expression for the resonating frequency of an LCR series resonating circuit.

Derive an expression for the impudence of a series LCR circuit connected to an ac supply of variable frequency. Plot a graph showing variation of current with the frequency of the applied voltage. Explain briefly how the phenomenon of resonance in the circuit can be used in the tuning mechanism of a radio or a TV set.

Show, with explanation, the variations of emf and current with time in $\mathrm{AC}$ circuits having pure resistance, pure inductance and pure capacitance.

 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?