A uniform magnetic field $B$ exists in the region between $x=0$ and $x=\frac{3R}{2}$ (region $2$ in the figure) pointing normally into the plane of the paper. A particle with charge $+Q$ and momentum $P$ directed along $x$-axis enters region $2$ from region $1$ at point $P_1(y=-R)$. If the particle enters into region $3$ through the point $P_2$ on the axis, then magnetic field $B$ is given as $\frac{8}{N}\frac{P}{QR}$, value of $N$ is _______

An arc of a circular loop of radius $R$ is kept in a plane, perpendicular to a uniform magnetic field $B$ as shown in figure. If the arc carries current $I$ then the magnetic force (in $N$) on the arc is
(Given $IRB=4\sqrt{2} \mathrm{N}$)

Three infinitely long current carrying wires are placed in a plane as shown in figure. The magnetic force (in $N$) on unit length of wire- $2$ is
(Given $\frac{{\mu }_0{I}_0^2}{d}=16\pi$ in $\mathrm{SI}$ unit)

Two current carrying circular loops are placed closely as shown in figure. If integral $\int \overrightarrow{B}·\overrightarrow{dr}$ along axis of the loops within range $-\infty$ to $\infty$ is $N$ times ${\mu }_{0}I$, then value of $N$ is _______

Six parallel infinite wires are placed at the vertices of a regular hexagon. Five wires carry current $I_0$ each while the sixth wire carries current $4I_0$ as shown. The resultant magnetic field at the centre of the hexagon is

A charged particle $(q, m)$ is projected with velocity $v_0\hat{i}$ towards a circular region of uniform magnetic field $B$ perpendicular and into the plane of paper as shown in the figure. $C$ is the centre of the circular region. If the angle between the line $CA$ and the $x-$ axis is $37°$ and the particle passes through point $C$ then the radius of the circular region $R$ is

The figure shows the path of a proton that passes through two regions containing uniform magnetic fields of magnitude $B_1$ and $B_2.$ Its path in each region is a half circle. Then

A current / flows in a long straight wire with crosssection having the form of a thin quarter-ring of radius $R$ (see figure). The magnetic field $\overrightarrow{B}$ at point $O$ if the current flows out of the page is