$A$ circular arc of radius $r$ carrying current $I$ subtends an angle $\frac{\pi}{16}$ at its centre. The radius of the metal wire is uniform. The magnetic induction at the centre of the circular arc is [where $\mu_0$ is the permeability of free space].

The dimensions of the ratio of magnetic flux $(\phi)$ and permeability $(\mu)$ are

Give two different properties of a current-carrying coil and state Ampere's suggestion.

$A$ thin charged rod is bent into the shape of a small circle of radius $R$, the charge per unit length of the rod being $\lambda$. The circle is rotated about its axis with a time period $T$, and it is found that the magnetic field at a distance $d$ away $(d >> R)$ from the center and on the axis varies as $\frac{R^{m}}{d^{n}}$. The values of $m$ and $n$ respectively are:

$A$ current of $1.5 \, A$ is flowing through an equilateral triangle of side $9 \, cm$. The magnetic field at the centroid of the triangle is (Assume that the current is flowing in the clockwise direction.)

$A$ long curved conductor carries a current $I$. $A$ small current element of length $dl$ on the wire induces a magnetic field at a point away from the current element. If the position vector between the current element and the point is $\vec{r}$, making an angle $\theta$ with the current element, then the induced magnetic field density $d\vec{B}$ at the point is $(\mu_0 = \text{permeability of free space})$: