Consider a ring of charge. Suppose that the charge q is not distributed uniformly over the ring but that charge is distributed uniformly over half the circumference and charge is distributed uniformly over the other half. Let . We have to find (a) the component of the electric field at any point on the axis directed along the axis and compare it with the uniform case; (b) the component of the electric field at any point on the axis perpendicular to the axis and compare it with the uniform case.

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As shown in the figure, on the circumference of a ring of radius R divided into two equal parts charge is distributed uniformly on part and charge is distributed uniformly on the other part. We will calculate the electric field at point P on the axis of the ring. A coordinate system has been fixed as shown in the figure.

Linear charge densities on the two halves of the ring will be

The component of the electric field at point P on the axis directed along it that is along the X-direction will be

We note that Y-axis has been chosen so that it divides the charge distribution on the ring symmetrically.

We calculate the component of the electric field along the Y-axis due to charge distributions and separately.

First we calculate the electric field component along the Y-axis due to charge distribution . It will be

Similarly, we can calculate the electric field component along the Y-axis due to charge distribution . It will be

Therefore, will be

We note that if the charge distribution along the circumference of the ring was uniform the component will be zero.

By symmetry, we note that is zero.