Question

A thread carrying a uniform charge X per unit length has the configurations shown in Fig. 3.2 a and b. Assuming a curvature radius R to be considerably less than the length of the thread, find the magnitude of the electric field strength at the point O.

Answer 1

(a) The net electric field strength at the point O due to straight parts of the thread equals zero. For the curved part (arc) let us derive a general expression i.e. let us calculate the field strength at the centre of arc of radius R and linear charge density λ and which subtends angle θ_o at the centre.

From the symmetry the sought field strength will be directed along the bisector of the angle θ_o and is given by

In our problem θ₀ = π/2 thus the field strength due to the turned part at the point  which is also the sought result.

(b) Using the solution of 3.14 (a) , net field strength at O due to stright parts equals

and is directed vertically down. Now using the solution of 3.15

(a), field strength due to the given curved part (semi-circle) at the point O becomes  and is directed vertically upward. Hence the sought net field strengh becomes zero.