Two circular coils X (smaller) and Y (bigger), each having a single turn, carry equal currents in the same direction and subtend the same angle at point 'O' along the axis of the coil. The distances of the centers of coil to point 'O' are $d$ and $\left(\dfrac{d}{2}\right)$ for coil Y and X respectively. The radii of coils Y and X are $(2r)$ and $(r)$ respectively. The magnetic induction due to the bigger coil at point 'O' is $B_y$ and that due to smaller coil X at point 'O' is $B_x$. ($d \gg r$) The relation between $B_x$ and $B_y$ is

Two thin long parallel wires, $W_1$ and $W_2$ separated by distance '$a$', carry currents $i$ and $3i$ respectively in the same direction. The magnitude of the force per unit length exerted by wire $W_1$ on wire $W_2$ is

Two different coils have self-inductance $3L$ and $L$. The current in both the coils is increased at the same constant rate. At certain instant of time, the power given to the two coils is same. At that time there was current and voltage induced in the two coils. At the same instant, the ratio of energy stored in the first coil to that in the second coil is

Consider two coils in which a current in one coil carrying $6$ A causes the change in the flux in the second coil $12 \times 10^{-4}$ weber/turn. The second coil has $2000$ turns. The mutual inductance between the coils is