An infinitely long straight non-magnetic conducting wire of radius a carries a dc current $I$. The magnetic field $B$, at a distance $r (r<a)$ from axis of the wire is

Electrons emitted with negligible speed from an electron gun are accelerated through a potential difference $V$ along the X-axis. These electrons emerges from a narrow hole into a uniform magnetic field $B$ directed along this axis. However, some of the electrons emerging from the hole make slightly divergent angles as shown in figure. These paraxial electrons meet for second time on the x-axis at a distance $\sqrt{\frac{N{\pi }^{2}mV}{e{B}^2}}$. Then find value of $N$. (Neglect interaction between electrons)

At time $t_1$, an electron is sent along the positive direction of $x$-axis, through both an electric field $E$ and a magnetic field $B$ , with $E$ directed parallel to the $y$-axis. Graph gives the y-component $F_{\mathrm{net}},y$ of the net force on the electron due to the two fields, as a function of the electron's speed $V$ at time ${\mathrm{t}}_1$. Assuming $\mathrm{Bx} = 0$, magnitude of electric field $\mathrm{E}$ in $\mathrm{mN} {\mathrm{C}}^{-1}$ is $50\mathrm{X}$, then find $\mathrm{X}$. (Use $e=1.6\times {10}^{–19} \mathrm{C}$)

A non-conducting disc having uniform positive charge $Q$, is rotating about its axis with uniform angular velocity $\omega$. The magnetic field at the centre of the disc is