A particle of mass $m$ and charge $q$ is projected into a region having a perpendicular magnetic field $B$. Find the angle of deviation (figure) of the particle as it comes out of the magnetic field if the width $d$ of the region is very slightly smaller than

$\left(\mathrm{a}\right) \frac{mv}{qB}$ $\left(\mathrm{b}\right) \frac{mv}{\sqrt{2}qB}$ and $\left(\mathrm{c}\right) \frac{3mv}{qB}$

The figure shows a convex lens of focal length $10 \mathrm{cm}$ lying in a uniform magnetic field $B$ of magnitude $1.2 \mathrm{T}$ parallel to its principal axis. A particle having a charge $2.0\times {10}^{–3} \mathrm{C}$ and mass $2.0\times {10}^{–5} \mathrm{kg}$ is projected perpendicular to the plane of the diagram with a speed of $4.8 \mathrm{m} {\mathrm{s}}^{-1}$. The particle moves along a circle with its centre on the principal axis at a distance of $15 \mathrm{cm}$ from the lens. The axis of the lens and of the circle are the same. Show that the image of the particle goes along a circle and find the radius of that circle.

An electron gun $\mathrm{G}$ emits electron of energy $2 \mathrm{keV}$ travelling in the positive $x-$direction. The electrons are required to hit the spot $S$ where $\mathrm{GS}=0.1 \mathrm{m}$ and the line $\mathrm{GS}$ makes an angle of $60°$ with the $x-$axis, as shown in the figure. A uniform magnetic field $B$ parallel to $\mathrm{GS}$ exists in the region outside of electron gun. Find the minimum value of $\mathrm{B}$ needed to make the electron hit $\mathrm{S}$. [Take mass of electron $= 9 \times {10}^{–31} \mathrm{kg}$]

A conducting wire of length $\mathcal{l}$, lying normal to a magnetic field $B$, moves with a velocity $v$ as shown in figure (a). Find the average magnetic force on a free electron of the wire. (b) Due to this magnetic force, electrons concentrate at one end resulting in an electric field inside the wire. The redistribution stops when the electric force on the free electrons balances the magnetic force. Find the electric field developed inside the wire when the redistribution stops. (c) What potential difference is developed between the ends of the wire?