$A$ $1 \ m$ long metal rod $AB$ completes the circuit as shown in the figure. The area of the circuit is perpendicular to the magnetic field of $0.10 \ T$. If the resistance of the total circuit is $2 \ \Omega$, then the force needed to move the rod towards the right with a constant speed $(v)$ of $1.5 \ m/s$ is . . . . . . $N$.

$A$ rod closing the circuit shown in the figure moves along a $U$-shaped wire at a constant speed $v$ under the action of a force $F$. The circuit is in a uniform magnetic field perpendicular to the plane. Calculate $F$ if the rate of heat generation in the circuit is $Q$.

In an $AC$ generator, if a coil of $N$ turns and area $A$ is rotated at $v$ revolutions per second in a uniform magnetic field $B$, then the motional $EMF$ produced is equal to (At $t=0$ $s$, the coil is perpendicular to the field).

$A$ conducting wire of parabolic shape,initially $y=x^2$,is moving with velocity $\vec{V} = V_0 \hat{i}$ in a non-uniform magnetic field $\vec{B} = B_0 \left(1 + \left(\frac{y}{L}\right)^\beta\right) \hat{k}$,as shown in the figure. If $V_0, B_0, L$ and $\beta$ are positive constants and $\Delta \phi$ is the potential difference developed between the ends of the wire,then the correct statement$(s)$ is/are:
$(1)$ $|\Delta \phi|$ remains the same if the parabolic wire is replaced by a straight wire,$y=x$ initially,of length $\sqrt{2} L$.
$(2)$ $|\Delta \phi|$ is proportional to the length of the wire projected on the $y$-axis.
$(3)$ $|\Delta \phi| = \frac{1}{2} B_0 V_0 L$ for $\beta = 0$.
$(4)$ $|\Delta \phi| = \frac{4}{3} B_0 V_0 L$ for $\beta = 2$.

$A$ conducting metal circular wire loop of radius $r$ is placed perpendicular to a magnetic field which varies with time as $B = B_0 e^{-t/\tau}$,where $B_0$ and $\tau$ are constants. If the resistance of the loop is $R$,then the total heat generated in the loop after a long time $(t \to \infty)$ is:

$A$ current carrying circular loop is perpendicular to a magnetic field of induction $10^{-4} \, T$. If the radius of the loop starts shrinking at a uniform rate of $2 \, mm/s$, then the emf induced in the loop at the instant, when its radius is $20 \, cm$ will be (in $\pi \, \mu V$)