1.
In a series resonant $$LCR$$ circuit, the voltage across $$R$$ is 100 volts and $$R = 1k\Omega $$ with $$C = 2\mu F.$$ The resonant frequency $$\omega $$ is $$200 rad/s.$$ At resonance the voltage across $$L$$ is
Across resistor, $$I = \frac{V}{R} = \frac{{100}}{{1000}} = 0.1A$$
At resonance, $${X_L} = {X_C} = \frac{1}{{\omega C}} = \frac{1}{{200 \times 2 \times {{10}^{ - 6}}}} = 2500$$
Voltage across $$L$$ is $$I\,{X_L} = 0.1 \times 2500 = 250V$$
2.
An arc lamp requires a direct current of $$10\,A$$ at $$80\,V$$ to function. If it is connected to a $$200\,V\left( {rms} \right),50\,Hz\,AC$$ supply, the series inductor needed for it to work is close to:
Here
$$\eqalign{
& i = \frac{e}{{\sqrt {{R^2} + X_L^2} }} = \frac{e}{{\sqrt {{R^2} + {\omega ^2}{L^2}} }} = \frac{e}{{\sqrt {{R^2} + 4{\pi ^2}{v^2}{L^2}} }} \cr
& 10 = \frac{{220}}{{\sqrt {64 + 4{\pi ^2}{{\left( {50} \right)}^2}L} }}\,\,\left[ {\because R = \frac{V}{I} = \frac{{80}}{{10}} = 8} \right] \cr} $$
On solving we get $$L= 0.065\,H$$
3.
A step down transformer is connected to 2400 volts line and 80 amperes of current is found to flow in output load. The ratio of the turns in primary and secondary coil is $$20 : 1.$$ If transformer efficiency is $$100\% ,$$ then the current flowing in the primary coil will be
4.
A resistor and an inductor are connected to an $$ac$$ supply of $$120\,V$$ and $$50\,Hz.$$ The current in the circuit is $$3\,A.$$ If the power consumed in the circuit is $$108\,W,$$ then the resistance in the circuit is
A.
$$12\,\Omega $$
B.
$$40\,\Omega $$
C.
$$\sqrt {\left( {52 \times 25} \right)} \,\Omega $$
In an $$ac$$ circuit, a pure indcutor does not consume any power. Therefore, power is consumed by the resistor only.
$$\eqalign{
& \therefore P = I_v^2R \cr
& {\text{or}}\,\,108 = {\left( 3 \right)^2}R\,\,{\text{or}}\,\,R = 12\,\Omega \cr} $$
5.
A wire of resistance $$R$$ is connected in series with an inductor of reactance $${\omega L}.$$ Then quality factor of $$RL$$ circuit is
A.
$$\frac{R}{{\omega L}}$$
B.
$$\frac{{\omega L}}{R}$$
C.
$$\frac{R}{{\sqrt {{R^2} + {\omega ^2}{L^2}} }}$$
D.
$$\frac{{\omega L}}{{\sqrt {{R^2} + {\omega ^2}{L^2}} }}$$
We define the quality factor of the circuit as follows
Quality factor, $$Q = 2\pi \times \frac{{{\text{total energy stored in the circuit}}}}{{{\text{loss in energy in each cycle}}}}$$
But the total energy stored in circuit $$ = Li_{rms}^2$$
and the energy loss per second $$ = i_{rms}^2R$$
So, loss in energy per cycle $$ = \frac{{i_{rms}^2R}}{f}$$
Hence, quality factor, $$Q = 2\pi \times \frac{{Li_{rms}^2}}{{\frac{{i_{rms}^2R}}{f}}}$$
$$ = \frac{{2\pi fL}}{R} = \frac{{\omega L}}{R}$$
6.
In an $$LCR$$ series $$a.c.$$ circuit, the voltage across each of the components, $$L, C$$ and $$R$$ is $$50V.$$ The voltage across the $$LC$$ combination will be
Since the phase difference between $$L\,\& \,C$$ is $$\pi ,$$
∴ net voltage difference across $$LC = 50 - 50 = 0$$
7.
An inductor $$20\,mH,$$ a capacitor $$50\,\mu F$$ and a resistor $$40\Omega $$ are connected in series across a source of emf $$V = 10\sin 340\,t.$$ The power loss in $$A.C.$$ circuit is :
8.
An inductor $$20\,mH,$$ a capacitor $$50\,\mu F$$ and a resistor $$40\,\Omega $$ are connected in series across a source of emf $$V = 10\sin 340\,t.$$ The power loss in $$AC$$ circuit is
Root mean square value of an alternating current is defined as the square root of the average of $${i^2},$$ during a complete cycle it may be taken by
$$\eqalign{
& \overline {{i^2}} = \frac{{\int_0^{\frac{{2\pi }}{\omega }} {{i^2}} dt}}{{\frac{{2\pi }}{\omega }}} \cr
& = \frac{{\int_0^{\frac{{2\pi }}{\omega }} {i_0^2} {{\sin }^2}\omega t\,dt}}{{\frac{{2\pi }}{\omega }}} \cr
& = \frac{{i_0^2\omega }}{{2\pi }}\int_0^{\frac{{2\pi }}{\omega }} {\frac{1}{2}} \left( {1 - \cos 2\omega t} \right)dt \cr
& = \frac{{i_0^2\omega }}{{4\pi }}\left[ {t - \frac{{\sin 2\omega t}}{{2\omega }}} \right]_0^{\frac{{2\pi }}{\omega }} \cr
& = \frac{{i_0^2\omega }}{{4\pi }}\left( {\frac{{2\pi }}{\omega }} \right) = \frac{{i_0^2}}{2} \cr
& \therefore {i_{rms}} = \sqrt {\overline {{i^2}} } = \frac{{{i_0}}}{{\sqrt 2 }} \cr} $$
10.
In a uniform magnetic field of induction $$B$$ a wire in the form of a semicircle of radius $$r$$ rotates about the diameter of the circle with an angular frequency $$\omega .$$ The axis of rotation is perpendicular to the field. If the total resistance of the circuit is $$R,$$ the mean power generated per period of rotation is
A.
$$\frac{{{{\left( {B\pi r\omega } \right)}^2}}}{{2R}}$$
B.
$$\frac{{{{\left( {B\pi {r^2}\omega } \right)}^2}}}{{8R}}$$
C.
$$\frac{{B\pi {r^2}\omega }}{{2R}}$$
D.
$$\frac{{{{\left( {B\pi r{\omega ^2}} \right)}^2}}}{{8R}}$$