At a distance l from a uniformly charged long wire, a charged particle is thrown radially outward with a velocity u in the direction perpendicular to the wire. When the particle reaches a distance 2l from the wire its speed is found to be `sqrt(2)u`. The magnitude of the velocity, when it is a distance 4l away from the wire, is (ignore gravity)

Options

A. `sqrt3u`
B. 2 u
C. `2sqrt2 u`
D. 4 u

Answer

A

Explanation

Energy conservation at A and B
`qV_A +1/2 "mu"^2 =qV_B+1/2 xx 2"mu"^2 implies q[V_A-V_B]=1/2"mu"^2`
`qxxlamda/(2piepsilon_0) ln 2 =1/2"mu"^2`
Energy conservation at A and C
`q[V_A-V_C]+1/2"mu"^2 =1/2 "mv"^2=1/2"mv"^2`
`implies (qlamda)/(2pi epsilon_0) ln 4 +1/2 "mu"^2 =1/2"mv^2`
`(2qlamda)/(2piepsilon_0)ln2 +1/2"mu"^2=1/2"mv"^2`
`implies 3/2u^2=1/2v^2 implies v=sqrt3u`