There is a line charge of infinite length with positive charge distributed uniformly along ita length. One electron is moving in a circular path with the plane of the circle perpendicular to the line charge.

Options

A. Kinetic energy of the electron is independent of its distance from line charge
B. Time taken by charged particle to complete one full circle around line charge is directly proportional to its distance from the line charge
C. Square of the time taken to complete one circle is proportional to the third power of its distance from the line charge.
D. Electric field intensity at a point due to line charge in inversely proportional to the length of perpendicular drawn from point on line charge.

Answer

A, B, D

Explanation

We know that the electric field intensity due to an infinite or long line charge at a perpendicular distance x from it is given by `E=lambda/(2piepsilon_0x)`. When electron uses this electric field to perform circular motion around the wire , we can write the following equation :
`exxlambda/(2piepsilon_0x)=(mv^2)/x rArr 1/2 mv^2 =(elambda)/(4piepsilon_0)`...(i)
In the above result we can see that the kinetic energy of the electron is independent of the distance of electron form line charge. Thus, option (a) is correct. From equation (i) we can write the following :
`v=sqrt((elambda)/(2pimepsilon_0))`
Time period of circular period of the electron is `(2pix)/v` and v is independent of distance. Hence, the time period is proportional to the distance of electron from the line charge or radius of its circular path . Option (b) is correct . If it is correct then option (c) cannot be correct.
In the expression of electric field intensity , `E=lambda/(2piepsilon_0x)`, x is perpendicular distance of point from the line charge , hence option (d) is correct.