An electron moves in a circular orbit with a uniform speed $v$. It produces a magnetic field $B$ at the centre of the circle. The radius of the circle is proportional to

$A$ particle carrying a charge equal to $100$ times the charge on an electron is rotating one rotation per second in a circular path of radius $0.8 \ m$. The value of magnetic field produced at the centre will be $(\mu_0 = \text{permeability of vacuum})$

$A$ coil of $n$ number of turns is wound tightly in the form of a spiral with inner and outer radii $a$ and $b$ respectively. When a current of strength $I$ is passed through the coil,the magnetic field at its centre is

In the current-carrying conductor $(AOCDEFG)$ as shown,the magnetic induction at the point $O$ is ($R_1$ and $R_2$ are radii of arcs $CD$ and $EF$ respectively,$I$ = current in the loop,$\mu_0$ = permeability of free space).

$A$ long solenoid with $1000 \, \text{turns/m}$ has a core material with relative permeability $500$ and volume $10^{3} \, \text{cm}^{3}$. If the core material is replaced by another material having relative permeability of $750$ with the same volume, maintaining the same current of $0.75 \, \text{A}$ in the solenoid, the fractional change in the magnetic moment of the core would be approximately $\left(\frac{x}{499}\right)$. Find the value of $x$.

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