Embibe Experts Solutions for Chapter: Alternating Current, Exercise 4: Exercise - 4

A box $P$ and a coil $\mathrm{Q}$ are connected in series with an $\mathrm{AC}$ source of variable frequency. The $\mathrm{EMF}$ of source is constant at $10 \mathrm{V}.$ Box $P$ contains a capacitance of $1 \mathrm{\mu F}$ in series with a resistance of $32 \mathrm{\Omega }.$ Coil $Q$ has a self inductance $4.9 \mathrm{mH}$ and a resistance of $68 \mathrm{\Omega }$. The frequency is adjusted so that the maximum current flows in $P$ and $Q$. Find the impedance of $P$ and $Q$ at this frequency. Also find the voltage across $P$ and $Q$, respectively.

In a series $\mathrm{LCR}$ circuit with an ac source of $50 \mathrm{V}, R=300\mathrm{\Omega },$ frequency $v=\frac{50}{\mathrm{\pi }} \mathrm{Hz}.$ The average electric field energy stored in the capacitor and average magnetic energy stored in the coil are $25 \mathrm{mJ} \mathrm{and} 5 \mathrm{mJ},$ respectively. The $\mathrm{RMS}$ current in the circuit is $0.10 \mathrm{A}$. Then find:

(a) Capacitance $\left(c\right)$ of capacitor 

(b) Inductance $\left(L\right)$ of inductor.

(c) The sum of rms potential difference across the three elements.

An inductor $20 \times {10}^{–3} \mathrm{Henry},$ a capacitor $100 \mathrm{\mu F}$ and a resistor $50 \mathrm{\Omega }$ are connected in series across a source of $\mathrm{EMF} V=\left(10 \mathrm{V}\right)\sin 314t.$ Find the energy dissipated in the circuit in $20 \min$. If resistance is removed from the circuit and the value of inductance is doubled, then find the variation of current with time ($t$ in second) in the new circuit.

The electric current in an $\mathrm{AC}$ circuit is given by $i=i_0\sin \omega t.$ What is the time taken by the current to change from its maximum value to the rms value?

A series $\mathrm{LCR}$ circuit with $L=0.125/\mathrm{\pi } \mathrm{H}, C=500/\mathrm{\pi } \mathrm{nF}, R=23 \mathrm{\Omega }$ is connected to a $230 \mathrm{V}$ variable frequency supply.

(a) What is the source frequency for which current amplitude is maximum? Obtain this maximum value.

(b) What is the source frequency for which average power absorbed by the circuit is maximum? Obtain the value of this maximum power.

(c) For what reactance of the circuit, the power transferred to the circuit is half the power at resonance? What is the current amplitude at this reactance?

(d) If $\mathrm{\omega }$ is the angular frequency at which the power consumed in the circuit is half the power at resonance, write an expression for $\mathrm{\omega }$.

(e) What is the $Q$-factor (quality factor) of the given circuit ?

The maximum values of the alternating voltages and current are $400 \mathrm{V} \mathrm{and} 20 \mathrm{A}$, respectively, in a circuit connected to $50 \mathrm{Hz}$supply and these quantities are sinusoidal. The instantaneous values of the voltage and current are $200 \sqrt{2} \mathrm{V} \mathrm{and} 10 \mathrm{A}$, respectively. At that instant both are increasing positively. Determine the average power consumed in the circuit. 

$\mathrm{A} 750 \mathrm{Hz}, 20 \mathrm{V}$ source is connected to a resistance of $100 \mathrm{\Omega },$ a capacitance of $1.0 \mathrm{\mu F}$ and an inductance of $0.18 \mathrm{H}$ in series. Calculate the following quantities: 

(a) Impedance of the circuit

(b) Draw an impedance diagram with suitable scale

(c) Power factor

(d) The time in which the resistance will get heated by $10º\mathrm{C},$ provided that the thermal capacity of resistance $= 2 \mathrm{J} º{\mathrm{C}}^{-1}$

In the given circuit 

Calculate

(a) Current in each branch.

(b) Power generated in each resistance

(c) Total power generated in the circuit.

(d) Net current drawn from source.

(e) Net impedance of the circuit.