The ratio of the magnetic field and the magnetic moment at the centre of a current-carrying circular loop is $x$. When both the current and the radius are doubled, the ratio will be:

An insulated wire is wound so that it forms a flat coil with $N = 200$ turns. The radius of the innermost turn is $r_1 = 3\text{ cm}$, and of the outermost turn $r_2 = 6\text{ cm}$. If $20\text{ mA}$ current flows in it, then the magnetic moment will be $\alpha \times 10^{-2}\text{ A.m}^2$. The value of $\alpha$ is . . . . . . .

The magnetic moment vectors $\mu_{s}$ and $\mu_{l}$ associated with the intrinsic spin angular momentum $S$ and orbital angular momentum $L$ respectively,of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by $\mu_{s} = -(e/m)S$ and $\mu_{l} = -(e/2m)L$. Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

Due to the flow of current in a circular loop of radius $R$,the magnetic field produced at the centre of the loop is $B$. The magnetic moment of the loop is

$A$ uniform conducting wire of length $12a$ and resistance $R$ is wound up as a current-carrying coil in the shape of:
$(i)$ an equilateral triangle of side $a$;
$(ii)$ a square of side $a$;
$(iii)$ a regular hexagon of side $a$.
The coil is connected to a voltage source $V_{0}$. Find the magnetic moment of the coils in each case.

$A$ coil of $n$ turns and radius $R$ carries a current $I$. It is unwound and rewound to make a new coil of radius $\frac{R}{3}$ and the same current is passed through it. The ratio of the magnetic moment of the new coil to that of the original coil is