Question

Define the linear charge density of an infinitely long thread, if the work force field to move a charge Q = 1 nC at a distance r ₁ = 5 cm to r ₂ = 2 cm in the direction perpendicular to the filament is 50 uJ.

Определить линейную плотность заряда бесконечно длинной нити, если работа сил поля по перемещению заряда Q = 1 нКл с расстояния r₁ = 5 cм до r₂ = 2 см в направлении перпендикулярном нити равна 50 мкДж.

Answer 1

I am not sure how much your book gives you.
I will assume that you know that:
E = [ 1/(2pieo) ] lambda/r
where lambda = charge per unit length
and
r is distance perpendicular to the line
then
Voltage potential of point a at ra with respect to potential at point b at rb is:
Va -Vb = integral from a to b of E dr
= [lambda/(2pieo)]integral dr/r
= [lambda/(2pieo)] ln (rb/ra)
calculate that
now
work done = charge* voltage difference