Magnetic Flux and Faraday's Law

Q.

A square loop of side 4 cm is lying on a horizontal table. A uniform magnetic field of 0.5 T is directed downwards at an angle of ${60}^o$ to the vertical as shown in Fig. 25.3. If the field increases from zero to its final value in 0.2 s, the emf induced in the loop will be

1. 1 mV

2. 2 mV

3. 3 mV

4. 4 mV

Q. A long straight wire is parallel to one edge of the loop as shown in the figure. If the current in the long wire varies with time as $I=I_0{e}^{-\frac{t}{\tau }}$, what will be the induced emf in the loop ?

1. $\frac{{\mu }_{0}bI}{\pi \tau }ln\left(\frac{d+a}{d}\right)$
2. $\frac{{\mu }_{0}bI}{2\pi \tau }ln\left(\frac{d+a}{a}\right)$
3. $\frac{2{\mu }_{0}bI}{\pi \tau }ln\left(\frac{d+a}{a}\right)$
4. $\frac{{\mu }_{0}bI}{\pi \tau }ln\left(\frac{d}{d+a}\right)$

Q. A coil of area $100{\text{cm}}^2$ has $500$ turns. Magnetc field of $0.1{\text{weber/m}}^2$ is perpendicular to the plane of coil. The magnetic field is reduced to zero in $0.1\text{s}$. Then induced e.m.f. in the coil is

1. $1\text{V}$
2. $5\text{V}$
3. $50\text{V}$
4. $0\text{V}$

Q. Figure below shows a conducting rod of negligible resistance that can slide on a smooth U-shaped rail made of a wire of resistance $1\Omega m^{-1}$. Position of the conducting rod at $t=0$ is shown. A time dependent magnetic field $B=2T\text{(tesla)}$ is switched ON at $t=0$.
At $t=0$, when the magnetic field is switched ON, the conducting rod is moved to the left at a constant speed of $5{\text{cms}}^{-1}$ by some external means. The rod moves perpendicular to the rail. At $t=2s$, induced emf has magnitude

1. 0.12 V
2. 0.08 V
3. 0.04 V
4. 0.02 V

Q. Square loop ABCD of area $20c{m}^2$ and resistance $5\Omega$ is rotated in a magnetic field B = 2T through ${180}^{\circ }$, with field perpendicualr to the loop initially. In 0.01 seconds the magnitude of induced e.m.f is …………….

1. 0.4 V
2. 0.8 V
3. 0.6 V
4. 1.0 V

Q. A part of circuit is shown in figure. Energy stored in $0.2\mu F$ capacitor in steady state in microjoules is

Q. Flux $\varphi$ (in $weber$) in a closed circuit of resistance $10\Omega$ varies with time $t$ (in $seconds$) according to the equation $\varphi =6t^2–5t+1$. What is the magnitude of the induced current at $t=0.25s$?

1. $1.2A$
2. $0.2A$
3. $0.6A$
4. $0.8A$

Q. A wire of length $l$, mass $m$ and resistance $R$ slides without any friction, down the parallel conducting rails of negligible resistance (see figure). The rails are connected to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and the rails form a closed rectangular conducting loop. The plane of the rails makes an angle $\theta$ with the horizontal and a uniform vertical magnetic field of induction B exists throughout the region. Find the steady state velocity of the wire.

1. $\frac{mg}{R}\frac{\sin \theta }{B^2{l}^2{\cos }^2\theta }$
2. $\frac{mg}{R}\frac{{\sin }^2\theta }{B^2{l}^2{\cos }^2\theta }$
3. $\frac{mgR\sin \theta }{B^2{l}^2{\cos }^2\theta }$
4. $mgR\frac{{\sin }^2\theta }{B^2{l}^2\cos \theta }$

Q. In the given figure, O is the origin of a cylindrical region having magnetic field $B=B_{0}t$. OAB is an equilateral triangular loop of side L. Radius of the cylinder is R.

1. EMF induced in side OA is $\frac{B_0\pi R^2}{18}$
2. EMF induced in side OB is zero
3. EMF induced in side AB is $\frac{B_0\pi R^2}{6}$
4. EMF induced in side AB is $\frac{B_0\pi R^2}{18}$

Q. In a very long solenoid of radius $R$, if the magnetic field changes at the rate of $\frac{dB}{dt}$, the induced emf for the triangular circuit ABC as shown in the figure is (AB=BC)

1. $R^2\left(\frac{dB}{dt}\right)$
2. $4R^2\left(\frac{dB}{dt}\right)$
3. $\frac{1}{2}{R}^2\left(\frac{dB}{dt}\right)$
4. $2R^2\left(\frac{dB}{dt}\right)$

Q. In the following figure, the magnet is moved towards the coil with a speed v and induced emf is e. If magnet and coil recede away from one another each moving with speed v, the induced emf in the coil will be

1. e
2. 2e
3. e/2
4. 4e

Q. A long straight wire is parallel to one edge as shown in figure. If the current in the long wire varies in time as $i=I_0{e}^{-t/\tau }$, what will be the induced emf in the loop ?

1. $\frac{{\mu }_{0}bi}{\pi \tau }ln\left(\frac{d+a}{d}\right)$
2. $\frac{{\mu }_{0}bi}{2\pi \tau }ln\left(\frac{d+a}{d}\right)$
3. $\frac{2{\mu }_{0}bi}{\pi \tau }ln\left(\frac{d+a}{d}\right)$
4. $\frac{{\mu }_{0}bi}{\pi \tau }ln\left(\frac{d}{d+a}\right)$

Q. A square wire loop of $10\text{cm}$ side lies at right angles to a uniform magnetic field of $20\text{T}$ in vertically downwards direction. A $10\text{V}$ light bulb is in a series with the loop as shown in the fig. The magnetic field is decreasing steadily to zero over a time interval $\Delta t$. The bulb will shine with full brightness if $\Delta t$ is equal to

1. $20\text{ms}$
2. $0.02\text{ms}$
3. $2\text{ms}$
4. $0.2\text{ms}$

Q. A long solenoid has $1000$ turns per meter and its radius is $R=2.5\text{cm}$. Current in the solenoid increases at a rate of $\frac{200}{\pi }\text{A/s}$. Find the magnitude of induced electric field at a point outside the solenoid at a distance $4\text{cm}$ from its axis.

1. $3\times {10}^{-4}\text{V/m}$
2. $6.25\times {10}^{-4}\text{V/m}$
3. $4.5\times {10}^{-4}\text{V/m}$
4. $9.25\times {10}^{-4}\text{V/m}$

Q. A long cylinderical magnet is moving at a constant velocity $v$ towards a circular wire ring whose radius is slightly larger than that of magnet and passes through it. Graph of flux vs time can be.

1. 
2. 
3. 
4. 

Q. A magnetic field B = 2t T exist in a cylinderical region as shown in figure. A square loop of side l = 2cm is placed inside the cylinder; such that, one of its vertices is at center of the cylinder. EMF induced in side AB of the square is

1. 0.8 mV
2. 0 V
3. 0.2 mV
4. 0.4 mV

Q. A coil of square shape $(4m\times 4m)$ is placed in magnetic filed as shown in the figure. Find induced emf in loop.

1. 32 volt
2. 8 volt
3. 16 volt
4. zero

Q. In the given figure, O is the origin of a cylindrical region having magnetic field $B=B_{0}t$. OAB is an equilateral triangular loop of side L. Radius of the cylinder is R.

1. EMF induced in side OA is $\frac{B_0\pi R^2}{18}$
2. EMF induced in side OB is zero
3. EMF induced in side AB is $\frac{B_0\pi R^2}{6}$
4. EMF induced in side AB is $\frac{B_0\pi R^2}{18}$

Q. A uniform but time varying magnetic field is present in a circular region of radius $R$. The magnetic field is perpendicular and into the plane of the field, and is increasing at a constant rate $\alpha$. There is a straight conducting rod of length $2R$ placed as shown in the figure. The magnitude of induced emf across the rod is given by $\frac{\pi R^2\alpha }{x}$. Find $x$.

Q.

A transformer is used to convert $200VACto11VAC$ .Number of turns in the primary of transformer is $2000.$How many turns are there in its secondary? if the efficiency of the transformer is $90\%$and it can work at maximum load of $10W.$ Calculate output power and output current.

Q. Figure below shows a conducting rod of negligible resistance that can slide on a smooth U-shaped rail made of a wire of resistance $1\Omega m^{-1}$. Position of the conducting rod at $t=0$ is shown. A time dependent magnetic field $B=2T\text{(tesla)}$ is switched ON at $t=0$.

The current in the loop at $t=0$ due to induced emf is

1. 0.16 A, clockwise
2. 0.08 A, clockwise
3. 0.08 A, anticlockwise
4. Zero

Q. A uniform but time varying magnetic field is present in a circular region of radius $R$. The magnetic field is perpendicular and into the plane of the field, and is increasing at a constant rate $\alpha$. There is a straight conducting rod of length $2R$ placed as shown in the figure. The magnitude of induced emf across the rod is given by $\frac{\pi R^2\alpha }{x}$. Find $x$.

Q. $P$ and $Q$ are two thin circular coils of same radius and subjected to the same rate of change of flux. The magnetic field is perpendicularly inwards to their plane and varrying with time. If coil $P$ is made of copper and $Q$ is made of iron, then the wrong statement is $\left({\rho }_{iron>{\rho }_{copper}}\right)$

1. emf induced in the two is the same.
2. the induced current in $P$ is more than that in $Q$.
3. Resistance of two coils are different.
4. the induced currents are the same in both the coils.

Q.
A highly conducting ring of radius R is perpendicular to and concentric with the axis of a long solenoid as shown in fig. The ring has a narrow gap of width d in its circumference. The solenoid has cross sectional area A and a uniform internal field of magnitude $B_0$. Now beginning at t = 0, the solenoid current is steadily increased so that the field magnitude at any time t is given by $B\left(t\right)=B_0+\alpha twhere\alpha >0$ . Assuming that no charge can flow across the gap, the end of ring which has excess of positive charge and the magnitude of induced e.m.f. in the ring are respectively

1. $X,A\alpha$
2. $X,\pi R^2\alpha$
3. $Y,\pi A^2\alpha$
4. $Y,\pi R^2\alpha$

Q. Figure shows two parallel and coaxial loops. The smaller loop (radius $r$) is above the larger loop (radius $R$), by distance $x$ ($x>>R$). The magnetic fielde due to current $i$ in the larger loop is nearly constant throughout the smaller loop. Suppose that $x$ is increasing at a constant rate of $\frac{dx}{dt}=v$. The induced emf in the smaller loop is

1. $\frac{{\mu }_0\pi i{R}^2{r}^{2}v}{x^4}$
2. $\frac{{\mu }_0\pi i{R}^2{r}^{2}v}{2x^4}$
3. $\frac{3{\mu }_0\pi i{R}^2{r}^{2}v}{2x^4}$
4. $\frac{{\mu }_0\pi i{R}^2{r}^{2}v}{3x^4}$

Q. Three capacitors of capcitances $3\mu \text{F}$, $4\mu \text{F}$ and $6\mu \text{F}$ are charged upto a potential difference of $2V,3V$ and $4V$ respectively. If terminal $\text{a}$ is connected with $\text{f,}$ terminal $\text{e}$ is connected with $\text{d}$ and $\text{c}$ is connected with $\text{b}$ then find the charge flowing through the circuit?

1. $24\mu \text{C}$
2. $9\mu \text{C}$
   
3. $16\mu \text{C}$
   
4. $12\mu \text{C}$

Q. Figure below shows a conducting rod of negligible resistance that can slide on a smooth U-shaped rail made of a wire of resistance $1\Omega m^{-1}$. Position of the conducting rod at $t=0$ is shown. A time dependent magnetic field $B=2T\text{(tesla)}$ is switched ON at $t=0$.
At $t=0$, when the magnetic field is switched ON, the conducting rod is moved to the left at a constant speed of $5{\text{cms}}^{-1}$ by some external means. The rod moves perpendicular to the rail. At $t=2s$, induced emf has magnitude

1. 0.12 V
2. 0.08 V
3. 0.04 V
4. 0.02 V