Ampere's Circuital Law

A long straight metal rod has a very long hole of of radius '$a$' drilled parallel to the rod axis as shown in the figure. If the rod carries a current '$i$' find the value of magnetic induction on the axis of the hole, where $OC=c$

A coaxial cable is made up of two conductors. The inner conductor is solid and is of radius $R_1$ and the outer conductor is hollow of inner radius $R_2$ and outer radius $R_3$. The space between the conductors is filled with air. The inner and outer conductors are carrying currents of equal magnitudes and in opposite directions. Then the variation of magnetic field with distance from the axis is best plotted as

Two long conductors are arranged as shown above to form overlapping cylinders, each of radius $r$, whose centres are seprated by a distance $d$. Current of density $\mathrm{J}$ flows into the plane of the page along the shaded part along the shaded portion of the other, as shown. What are the magnitude and direction of the magnetic field at point $A$?

Two infinitely long linear conductors are arranged perpendicular to each other and are in mutually perpendicular planes as shown in figure. If $I_1=2 \mathrm{A}$ along the $y$-axis and $I_2=3 \mathrm{A}$ along $- z$- axis and $AP=AB=1 \mathrm{cm}$. The value of magnetic field strength $\overrightarrow{B}$ at $P$ is

Six wires of current $I_1=1\mathrm{A}, I_2=2\mathrm{A}, I_3=3\mathrm{A},I_4=1\mathrm{A},I_5=5\mathrm{A}$ and $I_6=4\mathrm{A}$ cut the page perpendicularly at the points $1,2,3,4$ and $6$ respectively as shown in the figure. Find the magnitude of the integral $\frac{\oint \overrightarrow{B}·d\overrightarrow{l}}{{\mu }_0}$ around the closed path.

A very long straight conductor has a circular cross-section of radius $R$ and carries a current density $\mathrm{J}$. Inside the conductor there is a cylindrical hole of radius a whose axis is parallel to the axis of the conductor and a distance $b$ from it. Let the $z$-axis be the axis of the conductor, and let axis of the hole be at $x=b$. Find the magnetic field (i) on the $x$-axis at $x=2R$ (ii) on the $y$-axis at $y=2R$

Intensity of magnetic field of the earth at a point inside a hollow iron box is :

A cylindrical conductor of radius $R$ carries a current along its length. The current density $J$, however, it is not uniform over the cross section of the conductor but is a function of the radius according to $J=br$, where $b$ is a constant. Find an expression for the magnetic field $B$. (i) at $r_1<R$ & (ii) at distance $r_2>R$, measured from the axis

A long straight wire carries a current along the $x$-axis. Consider the points $A\left(0,1,0\right), B\left(0,1,1\right), C\left(1,0,1\right)$ and $D\left(1,1,1\right)$. Which of the following pairs of points will have magnetic fields of the same magnitude-