$A$ current of $0.1\, A$ circulates around a coil of $100$ turns and having a radius equal to $5\, cm$. The magnetic field set up at the centre of the coil is $({\mu _0} = 4\pi \times {10^{ - 7}}\,T\cdot m/A)$.

Two concentric rings are kept in the same plane. The number of turns in each ring is $25$. Their radii are $50 \text{ cm}$ and $200 \text{ cm}$,and they carry electric currents of $0.1 \text{ A}$ and $0.2 \text{ A}$ respectively in mutually opposite directions. The magnitude of the magnetic field produced at their centre is . . . . . . $\text{T}$.

$A$ 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

$A$ circular coil of radius $R$ carries an electric current $I$. The magnetic field due to the coil at a point on the axis of the coil located at a distance $r$ from the centre of the coil,such that $r \gg R$,the magnetic field at that point is proportional to

$A$ coil having $N$ turns carries a current $I$ as shown in the figure. The magnetic field intensity at point $P$ is

$A$ coil having $2000$ turns is wound tightly in the form of a spiral with inner and outer radii $1 \,cm$ and $3 \,cm$, respectively. When a current $\frac{1}{\pi} \,mA$ passes through the coil, the magnetic field at the centre is calculated to be $K \ln 3 \times 10^{-6} \,T$. The value of $K$ is