RMS Value of Current in Sinusoidal AC

Q. In an $AC$ circuit, the current is given by $I=4\sin (100\pi t+{30}^{\circ })\text{A}$. The current becomes maximum for the first time $(\text{After}t=0)$ at $t$ equal to

1. $\frac{1}{200}\text{s}$
2. $\frac{1}{300}\text{s}$
3. $\frac{1}{50}\text{s}$
4. $\frac{1}{100}\text{s}$

Q. A small square loop of wire of side $l$ is placed inside a large square loop of side $L(l<<L).$ The loops are co-planar and their centres coincide. The mutual induction of the system is proportional to

1. $\frac{l}{L}$
2. $\frac{l^2}{L}$
3. $\frac{L}{l}$
   
4. $\frac{L^2}{l}$

Q. The peak voltage in a $220\text{V}$ AC source is -

Take $\sqrt{2}=1.414$

1. $220\text{V}$
2. $320\text{V}$
3. $311\text{V}$
4. $430\text{V}$

Q. In an AC circuit, the current is given by the equation, $i=2\sqrt{13}\sin \left(50\pi t\right)\text{A}$. Calculate the time taken to reach from zero $($at $t=0)$ to its first maximum value.

1. $5\text{ms}$
2. $10\text{ms}$
3. $20\text{ms}$
4. $100\text{ms}$

Q. The alternating current is given by $i=\{\sqrt{42}\sin \left(\frac{2\pi }{T}t\right)+10\}\text{A}$. The r.m.s value of this current is $\text{A}$

Q. The magnetic flux in a closed circuit, of resistance $20\Omega$, varies with time $\left(t\right)$ according to the equation $\varphi =7t^2-4t$. Where $\varphi$ is in $\text{webers}$ and $\text{t}$ is in $\text{seconds}$. The magnitude of the induced current at $t=0.25\text{sec}$ is,

1. $0.25\text{A}$
2. $0.025\text{A}$
3. $50\text{mA}$
4. $175\text{mA}$

Q.

Find the time required for a 50 Hz alternating current to change its value from zero to the rms value.

Q. For the circuit shown, the value of current at time $t=3.2\text{s}$ will be
$\text{(in A)}$.



[Voltage distribution $V\left(t\right)$ is shown by Figure $1$ and the circuit is shown in Figure $2$]

Q.

The instantaneous voltages at three terminals marked X, Y and Z are given by

$V_X=V_0\sin \omega t$

$V_Y=V_0\sin \left(\omega t+\frac{2\pi }{3}\right)$ and

$V_Z=V_0\sin \left({\omega }_0+\frac{4\pi }{3}\right)$

An ideal voltmeter is configured to read rms of the value of potential difference between its terminals. It is connected between points points X and Y and then between Y and Z. The reading (S) of the voltmeter will be

1. $(V_{XY}{)}^{rms}=V_o\sqrt{\frac{3}{2}}$
2. $(V_{YZ}{)}^{rms}=V_o\sqrt{\frac{1}{2}}$

3. $(V_{XY}{)}^{rms}=V_o$

4. Independent of the choice of the two terminals

Q. A $10\Omega$ resistance is connected across $220\text{V}–50\text{Hz}AC$ supply. The time taken by the current to change from its maximum value to the $RMS$ value is

1. $2.5\text{ms}$
2. $1.5\text{ms}$
3. $4.5\text{ms}$
4. $3.0\text{ms}$

Q. The magnetic flux through a stationary loop with resistance $R$ varies during an interval of time $T$ as $\varphi =at(T-t)$. The heat generated during this time neglecting the inductance of the loop will be

1. $\frac{a^2{T}^3}{3R}$
2. $\frac{a^2{T}^2}{3R}$
3. $\frac{a^{2}T}{3R}$
4. $\frac{a^3{T}^2}{3R}$

Q. Find the average value of current (in A) as shown graphically in the figure, from $t=0$ to $t=2s$.

Q.

The household supply of electricity is at 220 V (rms value) and 50 Hz. Find the peak voltage and the least possible time in which the voltage can change from the rms value to zero

Q. Determine the rms value of a semi-circular current wave which has a maximum value of a Ampere. The y axis denotes current in Ampere and x axis denotes time.

1. $\left(\frac{1}{\sqrt{2}}\right)a$
2. $\sqrt{\frac{3}{2}}a$
3. $\sqrt{\frac{2}{3}}a$
4. $\sqrt{\frac{1}{3}}a$

Q. The magnetic field through a single loop of wire having radius $12\text{cm}$ and resistance $8.5\Omega$ changes with time as shown in the figure. The magnetic field is perpendicular to the plane of the loop. Plot the induced current as a function of time.

1. 
2. 
3. 
4. 

Q. In an AC generator, a coil with $N$ turns, all of them has same area $A$ and total resistance $R$, rotates with an angular frequency $\omega$ in a magnetic field $B$. The peak value of current in the coil is-

1. $\frac{NBA\omega }{R}$
2. $\frac{NBA}{R}$
3. $\frac{NBA\omega \sin \omega t}{R}$
4. $\frac{NBA\omega \cos \omega t}{R}$

Q. A conducting circular loop of face area $2.5\times {10}^{-3}{\text{m}}^2$ is arranged with its plane perpendicular to a magnetic field which varies as $B=0.20\sin \left(50\pi t\right)$. Find the net charge flowing through the loop during the interval $t=0$ to $t=40\text{ms}$.

Assume resistance of the loop to be $10\Omega$.

1. $1\text{C}$
2. $0.5\text{C}$
3. $0.25\text{C}$
4. Zero

Q. An AC source is rated $220\text{V},50\text{Hz}$. The average voltage, is calculated in a time interval of $0.01\text{s}$. It

1. must be zero
2. is never zero
3. may be zero
4. must be infinite

Q. If a direct current of value $a$ ampere is superimposed on an alternating current $I=b\sin \omega t$ flowing through a wire, what is the effective value of the resulting current in the circuit?

1. $\sqrt{a^2+b^2}$
2. $\sqrt{a^2+\frac{b^2}{4}}$
3. $\sqrt{a^2+\frac{b^2}{2}}$
4. $\sqrt{\frac{a^2}{2}+b^2}$

Q. A circular coil of radius $5\text{cm}$, has $500$ turns of a conducting wire. The approximate value of the coefficient of self induction of the coil will be -

1. $25\text{mH}$
2. $5\text{mH}$
3. $0.5\text{mH}$
4. $2.5\text{mH}$

Q. An alternating voltage, $V=60\sin \left(\pi t\right)$ is applied across a $20\Omega$ resistor. What will be the reading of an AC ammeter, connected in series with the resistor?

1. $3\text{A}$
2. $\frac{3}{\sqrt{2}}\text{A}$
3. $1.5\text{A}$
4. The reading of ammeter will change with time.

Q. Two neighbouring coils $\text{A}$ and $\text{B}$ have a mutual inductance of $20\text{mH}$. The current flowing through $\text{A}$ is given by $i=3t^2-4t+6$. The induced emf at $t=2\text{s}$ in coil $\text{B}$ is-

1. $160\text{mV}$
2. $200\text{mV}$
3. $260\text{mV}$
4. $300\text{mV}$

Q. A closed coil having $50\text{turns}$, area $300{\text{cm}}^2$, is rotated from a position where its plane makes an angle of ${60}^{\circ }$ with a magnetic field of flux density $2.0\text{T}$ to a position perpendicular to the field in a time of $10\text{s}$. What is the average $\text{emf}$ induced in the coil?

Q. Two coils $\text{P}$ and $\text{Q}$ are separated by some distance. When a current of $3\text{A}$ flows through coil $\text{P}$, a magnetic flux of ${10}^{-3}\text{Wb}$ passes through $\text{Q}$. No current is passed through $\text{Q}.$ When no current passes through $\text{P}$ and a current of $2\text{A}$ passes through $\text{Q}.$ The flux through $\text{P}$ is-

1. $6.67\times {10}^{-4}\text{Wb}$
2. $3.67\times {10}^{-4}\text{Wb}$
3. $6.67\times {10}^{-3}\text{Wb}$
4. $3.67\times {10}^{-3}\text{Wb}$

Q. Calculate the inductance per unit length of a double tape line as shown in the figure. The tapes are separated by a distance $h$ which is considerably less than their width $b$.

1. $\frac{{\mu }_{0}h}{b}$
2. $\frac{{\mu }_{0}h}{2b}$
3. $\frac{2{\mu }_{0}h}{b}$
4. $\frac{\sqrt{2}{\mu }_{0}h}{b}$

Q. A conducting loop of area $5{\text{cm}}^2$ is placed in a magnetic field which varies sinusoidally with time as $B=0.2\sin 300t$. The normal to the coil makes an angle of ${60}^{\circ }$ with the field. The emf induced at $t=\frac{\pi }{900}\text{s}$, is

1. $7.5\times {10}^{-3}\text{V}$
2. $\text{Zero}$
3. $15\times {10}^{-3}\text{V}$
4. $20\times {10}^{-3}\text{V}$

Q. Consider a long wire carrying a time varying current $i=kt(k>0)$. A circular loop of radius $a$ and resistance $R$ is placed with its center at a distance $d$ from the wire $(a<<d)$. The induced current in the loop is

1. $\frac{{\mu }_0{a}^{2}k}{4dR}$
2. $\frac{{\mu }_0{d}^{2}k}{2aR}$
3. $\frac{{\mu }_0{a}^{2}k}{2dR}$
4. ${\mu }_0\frac{(a+d{)}^{2}k}{2aR}$

Q. An alternating current is given by the equation, $i=3\sqrt{2}\sin (100\pi t+\pi /4)$ and an alternating voltage is given by the equation, $E=220\sqrt{2}\sin (100\pi t+\pi /2)$. Calculate the phase difference between them.

1. $\pi /4$
2. $\pi /3$
3. $\pi /2$
4. $\pi /6$

Q. Magnetic flux $\varphi$ (in Webers) linked with a closed circuit of resisitance $100\Omega$ varies with time $t$ (in seconds) as $\varphi =5t^2-4t+1$. The induced current in the circuit at $t=0.2\text{sec}$ is

1. $5\text{mA}$
2. $10\text{mA}$
3. $20\text{mA}$
4. $1\text{A}$

Q.
A plane loop is shaped in the form as shown in figure with radii $a=20\text{cm}$ and $b=10\text{cm}$ and is placed in a uniform time varying magnetic field $B=B_0\sin \omega t$ , where $B_0=10\text{mT}$ and $\omega =100\text{rad/s}$. Find the amplitude of the current induced(in $A$) in the loop, if its resistance per unit length is equal to $50\times {10}^{-3}\frac{\Omega }{\text{m}}$. The inductance of the loop is negligible.