A positive point charge `q` of mass `m`, kept at a distance `x_(0)`(in the same plane) from a fixed very long straight current is projected normally away from it with speed `v`.Find the maximum separation between the wire and the particle.

Explanation

Since the magnetic field is not unifrom, the particle doesn't follws circular path but the speed `(v)` of the particle is constant. Here the magnetic field set-up by the staight current is along the negative `z`-axis, the initially velocity of the particle is along `x`-axis and the froce `F` is in this `x-y` plane. The force at time `t` after starting from point `P` is
`F=q(vxxB)`
or `F=q[(v_(x)hati+v_(y)hatj)xx((mu_(0)I)/(2pix)(-k))]=(mu_(0)qI)/(2pix)(-v_(y)+v_(x)hatj)`
So, `F_(x)=(-mu_(0))/(2pi)(qlv_(y))/x , therefore a_(x)=(-mu_(0)qlv_(y))/(2pimx)`
or `(v_(x)dv_(x))/(dx)=(mu_(0)qlv_(y))/(2pimx)`..(i)
But `v_(x)^(2)+v_(y)^(2)=v^(2)`
`therefore 2v_(x)dv_(x)+2v_(y)dv_(y)=0 or v_(x)dv_(x)=-v_(y)dv_(y)` ..(ii)
From Eqs. (i) and (ii)
`(2pim)/(mu_(0)qI)=dv_(y)=(dx)/x or (2pim)/(mu_(0)qI)underset(0)overset(v)intdv_(y)=underset(x_(0))oversettgtg(x)int(dx)/x or (2pim)/(mu_(0)qI)v=lnx/x_(0)`
`therefore x=x_(0)e^(2pimv//muql)`