$A$ straight wire is placed in a magnetic field that varies with distance $x$ from the origin as $\vec{B} = B_0 \left( 2 - \frac{x}{a} \right) \hat{k}$. The ends of the wire are at $(a, 0)$ and $(2a, 0)$ and it carries a current $I$ in the positive $x$-direction. If the force on the wire is $\vec{F} = IB_0 \left( \frac{ka}{2} \right) \hat{j}$, then the value of $k$ is:

$A$ straight horizontal conducting rod of length $0.45\; m$ and mass $60\; g$ is suspended by two vertical wires at its ends. $A$ current of $5.0\; A$ is set up in the rod through the wires.
$(a)$ What magnetic field should be set up normal to the conductor in order that the tension in the wires is zero?
$(b)$ What will be the total tension in the wires if the direction of current is reversed keeping the magnetic field same as before? (Ignore the mass of the wires.) $g = 9.8\; m s^{-2}.$

As shown in the figure, a wire is bent to form a $D$-shaped loop carrying current $I$, where the curved part is a semi-circle of radius $R$. The loop is placed in a uniform magnetic field $\overrightarrow{B}$, which is directed into the plane of the paper. The net magnetic force on the closed loop is:

$A$ $3.0 \; cm$ wire carrying a current of $10 \; A$ is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be $0.27 \; T$. What is the magnetic force on the wire?

$A$ conductor (shown in the figure) carrying constant current $I$ is kept in the $x-y$ plane in a uniform magnetic field $\vec{B}$. If $F$ is the magnitude of the total magnetic force acting on the conductor,then the correct statement$(s)$ is(are):

Two wires $2 \text{ mm}$ apart supply current to a $100 \text{ V}$,$1 \text{ kW}$ heater. The force per metre between the wires is (Given $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$)