An infinite long wire holds a charge density  on it.A finite wire of length L is placed normal to the long wire and has a charge q uniformly distributed over it.

The net force on the finite wire is _____

a)q[logL]/2₀L

b)2q[logL]/₀L

c)q[log2L]/₀L

d)none

The standard formula for the electric field at a perpendicular distance r from the infinite wire of linear charge density  is      

 / 2 ₀ r. The infinite wire of linear charge density has an electric field of cylindrical symmetry given by the diagram, fig.1

 

Consider the wire of length L, charge q & linear charge density q / L at a perpendicular distance of R from the infinite wire,fig. 2 

Consider an infinitesimally small segment dx on the wire of length L at a perpendicular distance of x from the infinite wire.

 

Charge on the segment dx = q / L . dx

Force on this segment 

= Electric field due to infinite wire at a distance x . charge on dx 

=  / 2 ₀ x  .   q / L . dx

 

Force on the entire wire of length L is

 

 

_{[R ]}^{[R + L]}   / 2 ₀ x .  q / L . dx        [At the beginning of the wire x = R; at the end of the wire x = R + L ]

 

q / 2 ₀ L  _{[R ]}^{[R + L]}    dx / x 

 

= q / 2 ₀ L  . [ log (R+L) - log R ]

 

= q / 2 ₀ L  . log ( 1 + L / R )

 

Therefore, the net force on the finite wire is

 

 

 

 

Cheers !!