The magnetic field at the centre $O$ due to the given structure is:

$A$ straight wire carrying current $I$ is bent into a semi-circular arc of radius $r$,as shown. The magnitude of the magnetic field at point $O$ due to the semi-circular arc is ($\mu_{0} =$ permeability of free space).

The magnetic field due to a square loop of side $a$ carrying a current $I$ at its centre is:

$A$ battery is connected between two points $A$ and $B$ on the circumference of a uniform conducting ring of radius $r$ and resistance $R$. One of the arcs $AB$ of the ring subtends an angle $\theta$ at the centre. The value of the magnetic induction at the centre due to the current in the ring is

The magnetic field due to a current-carrying circular coil on its axis at a distance of $\sqrt{2} \,d$ from the centre of the coil is $B$. If $d$ is the diameter of the coil,then the magnetic field at the centre of the coil is: (in $B$)

$A$ non-conducting thin disc of radius $R$ rotates about its axis with an angular velocity $\omega$. The surface charge density on the disc varies with the distance $r$ from the center as $\sigma(r)=\sigma_0\left[1+\left(\frac{r}{R}\right)^\beta\right]$,where $\sigma_0$ and $\beta$ are constants. If the magnetic induction at the center is $B=\left(\frac{9}{10}\right) \mu_0 \sigma_0 \omega R$,the value of $\beta$ is