An arrangement with a pair of quarter circular coils of radii $r$ and $R$ with a common centre $C$ and carrying a current $I$ is shown in the figure. The permeability of free space is $\mu_0$. The magnetic field at $C$ is

A closely packed coil having $1000$ turns has an average radius of $62.8\,cm$. If current carried by the wire of the coil is $1\,A$, the value of magnetic field produced at the centre of the coil will be (permeability of free space $=4 \pi \times 10^{-7}\,H / m$ ) nearly

A length $L$ of wire carries a steady current $I$. It is bent first to form a circular plane coil of one turn. The same length is now bent more sharply to give a double loop of smaller radius. The magnetic field at the centre caused by the same current is

In a region of space, a uniform magnetic field $B$ exists in the $y-$direction.Aproton is fired from the origin, with its initial velocity $v$ making a small angle $\alpha$ with the $y-$ direction in the $yz$ plane. In the subsequent motion of the proton,

A circular current carrying coil has a radius $R$. The distance from the centre of the coil on the axis where the magnetic induction will be $\frac{1}{8}^{th}$ to its value at the centre of the coil, is

A circular loop is kept in that vertical plane which contains the north-south direction. It carries a current that is towards south at the topmost point. Let $A$ be a point on axis of the circle to the east of it and $B$ a point on this axis to the west of it. The magnetic field due to the loop :-