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 $\mathrm{x}$-axis enters region $2$ from region $1$ at point $P_1 (y = –R).$ Which of the following option(s) is/are correct ?

 

In the $xy$-plane, the region $y>0$ has a uniform magnetic field $B_1\widehat{ \mathrm{k}}$ and the region $y<0$ has another uniform magnetic field $B_2 \widehat{ \mathrm{k}}$. A positively charged particle is projected from the origin along the positive $y-$axis with speed $v_0=\pi {\mathrm{ms}}^{-1} \text{at} t=0$, as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the$x-$axis from below for the first time. If $B_2=4B_1$, the average speed of the particle, in ${\mathrm{ms}}^{-1}$, along the$x-$axis in the time interval $T$ is __________.

Two identical conducting wires $\mathrm{AOB}$and $\mathrm{COD}$ are placed at right angles to each other. The wire $\mathrm{AOB}$carries an electric current $I_1$ and $\mathrm{COD}$ carries a current $I_2$. The magnetic field on a point lying at a distance $d$ from $\mathrm{O}$, in a direction perpendicular to the plane of the wires $\mathrm{AOB}$ and $\mathrm{COD}$, will be given by