$A$ single current-carrying loop of wire with current $I$ flowing in an anticlockwise direction when seen from the $+ve\;z$ direction,lying in the $xy$ plane,is shown in the figure. The plot of the $\hat{j}$ component of the magnetic field $(B_y)$ at a distance $a$ (less than the radius of the coil) on the $yz$ plane versus the $z$ coordinate looks like:

Two similar coils of radius $R$ are lying concentrically with their planes at right angles to each other. The currents flowing in them are $I$ and $2I$,respectively. The resultant magnetic field induction at the centre will be

$A$ 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 $x$.

The unit of magnetic permeability is

Two infinitely long wires are placed at $(1 \text{ cm}, 1 \text{ cm})$ and $(1 \text{ cm}, -1 \text{ cm})$ with $1 \text{ A}$ current in each and in the same directions perpendicular to the $xy$-plane. Let the magnetic field due to these current-carrying wires at the origin be $B$. If $B_0$ is the magnitude of the field if only one of them was present,then $\frac{|B|}{B_0}$ is

$A$ long straight wire carrying a current of $30\,A$ is placed in an external uniform magnetic field of induction $4 \times 10^{-4}\,T$. The magnetic field is acting parallel to the direction of the current. The magnitude of the resultant magnetic induction in tesla at a point $2.0\,cm$ away from the wire is