An infinitely long wire carrying current $I$ is along the $Y$-axis such that its one end is at point $A(0, b)$ while the wire extends up to $+\infty$. Find the magnitude of the magnetic field strength at point $(a, 0)$.

The figure below shows three circuits consisting of concentric circular arcs and straight radial lines. The center of the circle is shown by the dot. The same current flows through each of the circuits. If $B_1, B_2, B_3$ are the magnitudes of the magnetic field at the center, which of the following is true?

Two circular loops having same radius $[ R = 10 \, cm ]$ and same current $[ I = \frac{7}{2} \, A ]$ are placed along the same axis as shown. If the distance between their centers is $[ 10 \, cm ]$,find the net magnetic field at point $P$ located at the midpoint between them.

Two parallel wires in the plane of the paper are at a distance $X_0$ apart. $A$ point charge is moving with speed $u$ between the wires in the same plane at a distance $X_1$ from one of the wires. When the wires carry current of magnitude $I$ in the same direction, the radius of curvature of the path of the point charge is $R_1$. In contrast, if the currents $I$ in the two wires have directions opposite to each other, the radius of curvature of the path is $R_2$. If $\frac{X_0}{X_1}=3$, the value of $\frac{R_1}{R_2}$ is:

$A$ negative charge is given to a non-conducting loop and the loop is rotated in the plane of paper about its centre as shown in the figure. The magnetic field produced by the ring affects a small magnet placed above the ring in the same plane.

$A$ 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 $x$.