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A thin nonconducting ring of radius R has a linear charge density λ λ0 cos φ, where λ0 is a constant, φ is the azimuthal angle. Find the magnitude of the electric field strength 

(a) at the centre of the ring; 

(b) on the axis of the ring as a function of the distance x from its centre. Investigate the obtained function at x >> R.

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(a) The given charge distribution is shown in Fig. The symmetry of this distribution implies that vector E at the point O is directed to the right, and its magnitude is equal to the sum of the projection onto the direction of vectors E of vector dE from elementary charges dq. The projection of vector dE onto vector E is

It should be noted that this integral is evaluated in the most simple way if we take into account that

(b) Take an element S at an azimuthal angle φ from the x-axis, the element subtending an angle dφ at the centre.

The elementary field at P due to the element is

The component alon OS can be broken into the parts along OX and OY with 

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