$A$ current flows through a circular coil of radius $R$. Find the distance from the center on the axis where the magnetic field is $\frac{1}{8}$th of the magnetic field at the center.

$A$ current carrying circular coil of radius $R$ produces magnetic field $B_1$ at an axial point $P$ at a distance $x$ from its centre and $B_2$ at point $Q$ placed at its centre respectively. If $B_2 = 8B_1$, the value of $x$ is

$A$ circular loop of radius $r$ is carrying current $I \ A$. The ratio of the magnetic field at the centre of the circular loop to the magnetic field at a distance $r$ from the centre of the loop on its axis is:

$A$ closely wound circular coil of radius $5\,cm$ produces a magnetic field of $37.68 \times 10^{-4}\,T$ at its center. The current through the coil is $......A$. [Given,number of turns in the coil is $100$ and $\pi=3.14$]

Write the proportionality constant of the Biot-Savart law with its unit and value (magnitude).

$A$ 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