A long wire having linear charge density `lambda` moving with constant velocity v along its length. A point charge moving with same speed in opposite direction and at that instant it is at a distance r from the wire. The net force acting on the charge is given by

Options

A. `(lambdaq)/(2pir)[(1)/(epsilon_(0))+v^(2)mu_(0)]`
B. `(lambdaq)/(2pir)[(1)/(epsilon_(0))-mu_(0)v^(2)]`
C. `(lambdaq)/(2pir)sqrt(((1)/(epsilon_(0)))^(2)+v^(4)mu_(0)^(2))`
D. zero

Explanation

Gauss Law for `vec(E)` inside sphere
`epsilon[E(4pir^(2))]=underset(0)overset(r )(int) (kr) (4pir^(2))dr`
`E=(kr^(2))/(2epsilon_(0))`
Electrostatic energy within `=underset(0)overset(R )(int)(1)/(2)epsilon_(0)E^(2)4pir^(2)dr=(2piepsilon_(0))underset(0)overset(R )(int)(k^(2)r^(6))/(16epsilon_(0)^(2))prop(R^(7))`
Total charge within `=underset(0)overset(R )(int)4pikr^(3)dr=pikR^(4)`
Electric field at `r gt R = (1)/(4piepsilon_(0))((pikR^(4))/(r^(2)))`
Electric energy for `oo gt r ge R = underset(R )overset(oo)(int)(((1)/(4piepsilon_(0))(pikR^(4))/(r^(2)))^(2)4pir^(2)dr)/(2epsilon_(0))=("const")R^(8)underset(R )overset(oo)(int)(1)/(r^(2))dr=("const")R^(8)[(1)/(R )]="const"(R^(7))`
Electrostatic energy inside and outside sphere is `propR^(7)`. So total `prop(R^(7))`