A long straight wire carries a current i. A particle having a positive charge q and mass m, kept at distance `x_0` from the wire is projected towards it with speed v. Find the closest distance of approach of charged particle to the wire.

Explanation

Magnetic field due to long wire is
`vecB=-(mu_0ihatk)/(2pix)`
Let the velocity of the charged particle at any instant be
`vecv=(v_xhati+v_yhatj)`
Magnetic force on the charge,
`vecF=q(vecvxxvecB)=-(mu_0qiv_y)/(2pix) hati+(mu_0qiv_x)/(2pix) hatj`
From Eq. (i), y component of acceleration is
`a_y=(dv_y)/(dt)=(mu_0qiv_x)/(2pimx)=(mu_0q)/(2pimx)i (dx)/(dt)`
At the minimum separation, velocity of particle is `-vhatj`
[closest distance will be when velocity becomes parallel to wire
or in -ve y-direction]
`int_0^-vdv_y=(mu_0qi)/(2piM)int_(x_0)^(x_(min))(dx)/x implies -v=(mu_0qi)/(2pim)In |(x_(min))/(x_0)|`
Thus, `x_(min)=x_0e^(-2pimv//mu_0qi)`