Faraday’s Law

The magnetic flux through each of five faces of a neutral playing dice is given ${\mathrm{\varphi }}_{\text{B}}=\pm \text{NWb}$, where N (= 1 to 5) is the number of spots on the face. The flux is positive (out-ward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die?

A jet plane is travelling towards wets at a speed of $1800 \mathrm{km} {\mathrm{h}}^{-1}$. What is the voltage difference developed between the ends of the wing having a span of $25 \mathrm{m}$, if the Earth’s magnetic field at the location has a magnitude of  $5\times {10}^{-4} \mathrm{T}$ and the dip angle is $30°$ ?

Figure shows a rectangular conductor $PQRS$ in which the conductor $PQ$ is free to move in a uniform magnetic field $B$ perpendicular to the plane of the paper. The field extends from $x=0$ to $x=b$ and is zero for $x>b$. Assume that only the arm $PQ$ possesses resistance $r$. When the arm $PQ$ is pulled outward from $x=0$ with constant speed $v$, the joules heating loss from $b\le x<2b$ would be:

According to Faraday’s law of electromagnetic induction:

A rectangular loop and a circular loop are moving out of a uniform magnetic field to a field – free region with a constant velocity ‘v’ as shown in the figure. Explain which loop do you expect the induced emf to be constant during the passage out of the field region. The magnetic field is normal to the loops.

The figure given below shows an arrangement by which current flows through the bulb (X) connected with coil B, when a.c. is passed through coil A.


(i) Name the phenomenon involved.

(ii) If a copper sheet is inserted in the gap between the coils, explain how the brightness of the bulb would change.

(i) In an a.c. generator, coil of $N$ turns and area $A$ is rotated at $v$ revolution per second in a uniform magnetic field $B$. Write the expression for the emf produced.

(ii) A $100$ turn coil of area  $0.1{\text{ m}}^{\text{2}}$  rotates at half a revolution per second. It is placed in a magnetic field $0.01 \mathrm{T}$ perpendicular to the axis of rotation of the coil. Calculate the maximum voltage generated in the coil.

A rectangular conducting loop of length $4 \mathrm{cm}$ and width $2 \mathrm{cm}$ is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2}\stackrel{\wedge }{x}+\frac{{\displaystyle 1}}{{\displaystyle 2}}\stackrel{\wedge }{y}$ with a constant speed $v$. The wire is carrying a steady current $I=10 \mathrm{A}$ in the positive $x$-direction. A current of $10 \mu \mathrm{A}$ flows through the loop when it is at a distance $d=4 \mathrm{cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$, then the value of $v$ is _____ $\mathrm{m} {\mathrm{s}}^{-1}$.

[Given: The permeability of free space ${\mu }_0=4\pi \times {10}^{-7} \mathrm{N} {\mathrm{A}}^{-2}$]

Two coaxial circular loops are shown in figure, smaller loop (radius $=r$) is a distance $x$ above the larger one (Radius $=R$) with $x>>R$. Now if $x$ is changing at a constant rate $\frac{dx}{dt}=v>0$ and current is flowing in the larger loop as shown then :

Two long parallel conducting rails of zero resistance separated by a distance $L$ are joined to a cell of emf $E$ at one end. An external uniform magnetic field $B$ is applied normal to the plane and into the plane of the rails as shown in the figure. A conducting bar of mass $m$ and resistance $R$ is placed across the rails. The bar can slide freely parallel to itself always remaining perpendicular to the rails.

Uniform but time varying magnetic field $\mathrm{B}=\frac{{\mathrm{B}}_0\mathrm{t}}{\mathrm{T}}$ is present in the cylindrical region. There is small section at corners which is made with insulating material and other portion of rectangular loop is conducting. Loop is moving with constant velocity $\mathrm{v}$. At $\mathrm{t}=\mathrm{T}$ situation is as shown in figure . $\mathrm{C}$ is centre of cylindrical region. Choose correct option/s at the given instant.

A metallic horizontal ring of mass $m$ and radius $r$ falling under gravity in a region having magnetic field. If $z$ is the vertical direction, the $z$-component of magnetic field is $B_0(1+az)$ where $B_0$ and a are constants. If $R$ is the resistance of the ring, the terminal velocity of the ring is

A $L$ shaped conducting rod $A B C$ is lying in $xy$-plane and moving with velocity $\overrightarrow{V}=V_0\hat{\mathrm{i}}+3V_0\hat{\mathrm{j}}$ as shown in the figure. There exist a non uniform magnetic field which depends on $x$ and $y$ co-ordinate as $\overrightarrow{B}=\frac{B_0}{L_0}\left(x+y\right)\left(-\hat{\mathrm{k}}\right)$, where $B_0$ and $L_0$ are positive constants with proper dimensions. If $V_P-V_Q$ represents the motional induced emf between point $P$ and $Q$. Then choose the CORRECT option(s).

A square loop of side a, resistance $R$, mass $m$ is sliding as shown on a smooth horizontal table with speed $v_0$. It enters a uniform magnetic field of magnitude $B_0$ perpendicular to the table. It is seen that the loop comes to rest after entering a distance $l$ inside the magnetic field. The value of $v_0$ can be :-

A rectangular frame $ABCD$ made of a uniform metal wire has a straight connection between $E$ and $F$ made of the same wire as shown in the figure. $AEFD$ is a square of side $1 \mathrm{m}$ and $EF=FC=0.5 \mathrm{m}$. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the plane of the paper and normal to it. The rate of change of the magnetic field is $1 \mathrm{T} {\mathrm{s}}^{-1}$, the resistance per unit length of the wire is $1 \mathrm{\Omega } {\mathrm{m}}^{-1}$.  If the current in segments $BE$ is $\frac{x}{22} \mathrm{A}$, find $x$.

As shown, a uniform magnetic field $B$ pointing out of paper plane is confined in the cylindrical region of cross-section radius r. At a distance $R$ from the center of shaded area there is point particle of mass m and carrying charge $q$. The magnetic field is then quickly changed to zero. The speed of particle just after magnetic field reduces to zero is?

Take: Initial magnetic field $B = 6T$
Charge $q = 80 mC$
Radius $r = 5\mathrm{cm}, R = 10\mathrm{cm}$
Mass of particle $= 2 \mathrm{gram}$.

A Cylindrical region of radius $1 \mathrm{m}$ has instantaneous homogeneous magnetic field of $5 \mathrm{T}$ and it is increasing at a rate of $2 \mathrm{T} {\mathrm{s}}^{-1}$. The regular hexagonal loop $ABCDEFA$ of side $1 \mathrm{m}$ is being drawn in to the region with a constant speed of $1 \mathrm{m} {\mathrm{s}}^{-1}$ as shown in the figure. What is the magnitude of emf developed in the loop just after the shown instant when the corner $A$ of the hexagon is coinciding with the centre of the circle?

Uniform but time varying magnetic field $B=\frac{B_{0}t}{T}$ present in cylindrical region of radius $R$. There is a circular conducting ring of radius a is kept at distance $5$ a (centre to centre distance). Choose the correct options :

A long straight wire carries a current $I$ along $y$-axis. The magnetic flux through a square placed a shown is $\frac{{\mu }_{0}I}{p\pi }ln\left(q\right)$. The $p\times q=$