The magnetic field at point $P$ in the hollow cylindrical wire carrying a current $i$ shown in the figure at a distance $r$ from axis $(a<r<b)$ is given by (Taking current density to be uniform across its cross-section)

A long straight wire is kept along $x-$axis. It carries a current $i$ in the positive $x$-direction. A proton and an electron are placed at $(0, a, 0)$ and $(0, -a, 0)$ respectively. The proton is imparted an initial velocity $v$ along $+z$ axis and the electron is imparted an initial velocity $v$ along $+x$-axis. The magnetic forces experienced by the two particles at the instant are

In the plane of coil a uniform magnetic field $B$ exists. Torque on the current carrying coil of radius $R$ is

Calculate the torque acting upon the following structure carrying current $/$ due to the magnetic field $B.$ Both the magnetic field and the structure are in the plane of paper as shown.

An infinite semi-cylindrical conducting shell of radius $R$ carrying a current $I$ along$-z$ direction as shown in figure. If a charge particle $(q, m)$ is projected from point $O$ (on the axis of semi-cylindrical shell) with velocity $\overrightarrow{v}=v_0\hat{j},$ then force on this charge particle will be

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A charge particle $\left(\frac{\mathrm{q}}{\mathrm{m}}=1 \mathrm{in} \mathrm{S}.\mathrm{I}. \mathrm{unit}\right)$ is projected at $t=0$ with velocity $5\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ at an angle of $45°$ with $+x$ direction as shown in a region where uniform magnetic field $\overrightarrow{B}=\left(-\frac{\hat{i}}{\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}}\right) T$  exist. Velocity of charge particle at $t=\frac{\pi }{4} \mathrm{s}$ will be

A charge particle $(q=1\mathrm{C}, m=1 \mathrm{kg})$ moving with velocity $\overrightarrow{v}=5\hat{i} \mathrm{m} {\mathrm{s}}^{-1}$ passes through a region of width $\left(D=\frac{3}{2} \mathrm{m}\right)$ where uniform magnetic field $\overrightarrow{B}=-2\hat{k} \mathrm{T}$ exist as shown. Find average force exerted by magnetic field on this charge particle.