Value of $r$ for which magnetic field at point $P$ on the axis of a circular current carrying loop of radius $a$ is maximum as shown in figure ______

Magnetic field at $O$ of a loop of uniform resistance per unit length is ______

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 _______