Find the relation between the magnetic field at $\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ on axis, and magnetic field at the center of coil. Here $R$ is radius of the coil.

A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of $30$ turns and radius $12 \mathrm{cm}$. The coil is in a vertical plane making an angle of $45°$ with the magnetic meridian. When the current in the coil is $0·35 \mathrm{A}$, the needle points west to east. Determine the horizontal component of the Earth's magnetic field at the location.
Next, the current in the coil is reversed and the coil is rotated about its vertical axis by an angle of $90°$ in the anti clockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the places to be zero.

The figure $\left(\mathrm{a}\right)$ shows two wires, each carrying a current. The wire $1$ consists of a circular arc of radius $R$ and two radial lengths; it carries current $i_1=2.0 \mathrm{A}$ in the direction indicated. The wire $2$ is long and straight; it carries a current $i_2$ that can be varied, and it is at a distance of $\frac{R}{2}$ from the Centre of the arc. The net magnetic field $\overrightarrow{B}$ due to the two currents is measured at the Centre of curvature of the arc. Figure $\left(\mathrm{b}\right)$ is a plot of the component of $\overrightarrow{B}$ in the direction perpendicular to the figure as a function of current $i_2$. What is the angle subtended by the arc?