Two concentric coplanar circular loops of radii $r_1$ and $r_2$ carry currents $i_1$ and $i_2$ in opposite directions (one clockwise and the other anticlockwise). The magnetic induction at the center of the loops is half that due to $i_1$ alone at the center. If $r_2 = 2r_1$,find the value of $\frac{i_2}{i_1}$.

An electron makes a full rotation in a circle of radius $0.8 \ m$ in one second. The magnetic field at the centre of the circle is $(\mu_0 = 4 \pi \times 10^{-7} \text{ SI units})$.

The Earth's magnetic induction at a certain point is $7 \times 10^{-5} \ Wb/m^2$. This is to be annulled by the magnetic induction at the center of a circular conducting loop of radius $5 \ cm$. The required current in the loop is......$A$

$A$ long solenoid with $1000 \, \text{turns/m}$ has a core material with relative permeability $500$ and volume $10^{3} \, \text{cm}^{3}$. If the core material is replaced by another material having relative permeability of $750$ with the same volume, maintaining the same current of $0.75 \, \text{A}$ in the solenoid, the fractional change in the magnetic moment of the core would be approximately $\left(\frac{x}{499}\right)$. Find the value of $x$.

The magnetic field at the origin due to a current element $i d l$ placed at a point with vector position $r$ is

Two concentric coils $X$ and $Y$ of radii $16 \, cm$ and $10 \, cm$ lie in the same vertical plane containing $N-S$ direction. Coil $X$ has $20$ turns and carries $16 \, A$. Coil $Y$ has $25$ turns and carries $18 \, A$. Coil $X$ has current in anticlockwise direction and coil $Y$ has current in clockwise direction for an observer looking at the coils facing the west. The magnitude of the net magnetic field at their common centre is: