$A$ wire of length $1\, m$ is perpendicular to the $x-y$ plane. It is moved with velocity $\vec{v} = (3\hat{i} + 3\hat{j} + 2\hat{k})\, m/s$ through a region of uniform magnetic field $\vec{B} = (\hat{i} + 2\hat{j})\, T$. The potential difference between the ends of the wire is (in $V$):

$A$ player with a $3 \text{ m}$ long iron rod runs towards the east with a speed of $30 \text{ km/hr}$. The horizontal component of the Earth's magnetic field is $4 \times 10^{-5} \text{ Wb/m}^2$. If the player is running with the rod in horizontal and vertical positions,then the potential difference induced between the two ends of the rod in the two cases will be:

$A$ 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:

When a $J$-shaped conducting rod is rotating in its own plane with constant angular velocity $\omega$,about one of its ends $P$,in a uniform magnetic field $\vec B$ directed normally into the plane of the paper,then the magnitude of the emf induced across it will be:

$A$ rectangular loop of length $l$ and breadth $b$ is placed at a distance of $x$ from an infinitely long wire carrying current $i$ such that the direction of the current is parallel to the breadth of the loop. If the loop moves away from the current-carrying wire in a direction perpendicular to it with a velocity $v$,the magnitude of the induced emf in the loop is: ($\mu_0=$ permeability of free space)

$A$ 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 $\beta = 2$.