Figure shows a cylindrical conductor of inner radius $a$ and outer radius $b$ which carries a current $i$ uniformly spread over its cross section. Show that the magnetic field $B$ for points inside the body of the conductor (i.e., $a<r<b$) is given by $B=\frac{{\mu }_{0}i}{2\pi \left(b^2-a^2\right)}\frac{r^2-a^2}{r}$. This formula for the limiting case of $a=0$.

A thin but long and hollow cylindrical tube of radius $r$ carries a current $i$ along its length. Find the magnitude of the magnetic field at a distance $r/2$ from the surface:

$\left(\mathrm{a}\right)$ inside the tube.

$\left(\mathrm{b}\right)$ outside the tube.

A tightly-wound, long solenoid is kept with its axis parallel to a large metal sheet carrying a surface current. The surface current through the width $dl$ of the sheet is $Kdl$ and the number of turns per unit length of the solenoid is $n$. The magnetic field near the centre of the solenoid is found to be zero,

$\left(\mathrm{a}\right)$ Find the current in the solenoid. $\left(\mathrm{b}\right)$ If the solenoid is rotated to make its axis perpendicular to the metal sheet, what would be the magnitude of the magnetic field near its centre?

Two large metal sheets carry surface currents as shown in figure. The current through a strip of width $dl$ is $Kdl$, where $K$ is a constant. Find the magnetic field at the point $P, Q$ and $R$.

Magnetic effect of the current and magnetic force on charge or current a neutron, a proton, an electron and an $\mathrm{\alpha }$-particle enters a uniform magnetic field with equal velocities. The field is directed along the inward normal to the plane of the paper. Which of these tracks followed are by electron and $\mathrm{\alpha }$-particle.

Two long parallel wires carrying currents $2.5 \stackrel{\circ }{\mathrm{A}}$ and $\mathrm{I} \stackrel{\circ }{\mathrm{A}}$ in the same direction (directed into the plane of the paper) are held at $P$ and $Q$, respectively, such that they are perpendicular to the plane of paper. The points $P$ and $Q$ are located at a distance of $5 \mathrm{m}$ and $2 \mathrm{m}$, respectively, from a collinear point $R$.

(a) An electron moving with a velocity of $4\times {10}^5 \mathrm{m} {\mathrm{s}}^{-1}$ along the positive $X$-direction experiences a force of magnitude $3.2\times {10}^{-20} \mathrm{N}$ at the point $R$. Find the value of $I$.

(b) Find all the positions at which a third long-parallel wire carrying a current of magnitude $2.5 \mathrm{A}$ may be placed so that the magnetic induction at $R$ is zero.