A thin nonconducting ring of radius `R` has a linear charge density `lambda = lambda_(0) cos varphi`, where `lambda_(0)` is a constant , `phi` is the azimutahl 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. Investegation the obtained function at `x gt gt R`.

Explanation

To solve the problem, we will break it down into two parts:

### Part (a): Electric Field at the Center of the Ring

1. **Understanding the Charge Distribution**: The linear charge density is given by \( \lambda(\varphi) = \lambda_0 \cos \varphi \). This means that the charge is distributed non-uniformly along the ring.

2. **Consider a Small Element of Charge**: Take a small element of the ring \( d\varphi \) at an angle \( \varphi \). The charge \( dq \) on this element is:
\[
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