A positive point charge is released from rest at a distance `r _ 0 ` from a positive line charge with uniform density. The speed (v) of the point charge, as a function of instantaneous distance r from line charge, is proportional to:

Options

A. `v prop e^(pi//tau_0)`
B. `v prop ln (r/r_0)`
C. `v prop sqrt(ln(r/r_0))`
D. `v prop (r/r_0)`

Answer

C

Explanation

Electric field due to line charge at a distance x can be written as : `E=lambda/(2piepsilon_0x)`
`qE=m(dv)/(dt) rArr (d lambda)/(2piepsilon_0x) =m(dv)/(dx)(dx)/(dt)`
`rArr m(dv)/(dx)v =(qlambda)/(2piepsilon_0x)rArr vdv=(qlambda)/(2piepsilon_0m) (dx)/x`
`rArr int_0^v vdv =(qlambda)/(2piepsilon_0m) int_(r_0)^(r)(dx)/x`
`rArr v^2/v =(qlambda)/(2piepsilon_0m) [ln x]_(r_0)^(r)=(qlambda)/(2piepsilon_0m) ln r/r_0`
`rArr v=sqrt((q lambda)/(piepsilon_0m)ln r/r_0)`
`rArr v prop sqrt(ln r/r_0)`