Magnetic Field Due to a Current Element, Biot-Savart Law

An electron and a positron are released from $(0, 0, 0)$ and $(0, 0, 1.5 R)$ respectively, in a uniform magnetic field $B=B_{\circ } \hat{\mathrm{i}}$, each with an equal momentum of magnitude $p=e B R$. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?

A $100$turn rectangular coil $\mathrm{ABCD}$(in $\mathrm{XY}$plane) is hung from one arm of a balance (Fig). A mass $500\mathrm{g}$is added to the other arm to balance the weight of the coil. A current $4.9 \mathrm{A}$ passes through the coil and a constant magnetic field of $0.2 \mathrm{T}$ acting inward (in xz plane) is switched on such that only arm $\mathrm{CD}$of length $1 \mathrm{cm}$ lies in the field. How much additional mass$‘m’$ must be added to regain the balance?

A current carrying circular loop of radius $R$ is placed in the $x-y$ plane with centre at the origin. Half of the loop with $x>0$ is now bent so that it now lies in the $y-z$ plane.
 

Biot-Savart law indicates that the moving electrons (velocity $v$) produce a magnetic field $\mathrm{B}$ such that