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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.

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(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 θo at the centre.

From the symmetry the sought field strength will be directed along the bisector of the angle θo 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.

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