In the cylindrical region of radius `R=2m`, there exists a time varying magnetic field `B` such that `(dB)/(dt)=2"Tesla"//sec`. A charged particle having charge `q=2C` is placed at the point `P` at a distance `d` from its center `O`. Now, the particle is moved in the direction perpendicular to line `OP` by an external agent up to infinity so that there is no gain in kinetic energy of charged particle. Then

Options

A. work done by external agent is `4pi` Joule when `d=4m`
B. work done by external agent is `4pi` Joule when `d=8m`
C. work done by external agent is independent of `d`
D. work done by external agen is positive.

Answer

A, B, C, D

Explanation

`intvecE=vec(dl)=A . (dB)/(dt)`
`E=2pisqrt(x^(2)+d^(2))=piR^(2)k`, where `((dB)/(dt)=k)`
`E=(R^(2)k)/(2sqrt(x^(2)+d^(2)))`
`W=int_(0)^(oo) dvecE.dvecx`
`=int_(0)^(oo) qEcostheta.dx=int_(0)^(oo) (qR^(2)k)/(2sqrt(x^(2)+x^(2)))d/(sqrt(x^(2)+x^(2)))dx=(qR^(2)kpi)/4`
Alternate solution
By applying Faraday's Law for an imaginary triangular loop having vertex `O,P` and point at infinity, `qintvecE.vec(dl)` can be found out.