The magnetic field due to a straight conductor of uniform cross-section of radius $a$ and carrying a steady current is represented by

$A$ 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 $x$.

$A$ square coil of side $a$ carries a current $I$. The magnetic field at the centre of the coil is

$A$ small current element of length $d\ell$ carrying current $I$ is placed at $(1, 1, 0)$ and is carrying current in the $+z$ direction. If the magnetic field at the origin is $\vec{B}_1$ and at the point $(2, 2, 0)$ is $\vec{B}_2$,then:

Three rings,each having equal radius $R$,are placed mutually perpendicular to each other,and each has its centre at the origin of the coordinate system. If current $I$ is flowing through each ring,then the magnitude of the magnetic field at the common centre is

It is found that a non-zero current element is unable to produce any magnetic field at a particular point. Then the angle between the current element and the position vector of that point with respect to the current element is