Electric Field due to a uniformly Charged Ring

Let's consider a uniformly charged ring of radius r and having a total charge +q. We are interested in finding the electric field strength at a point P lying on the ring's axis at a distance l from the centre of the ring.

Pick up an infinitesimally small arc element on the ring holding a small charge dq. As the size (or dimensions) of this element is too small compared to its separation from the point of calculation of field (the point P), we can approximate it as a point charge. Therefore the field produced by the considered arc element is d(as shown in the above figure) such that |dE| = k(dq)/R². Here R is the distance of point P from the arc element.

The charge distribution is symmetric with respect to the axis of the ring. Therefore at any point on the axis of the ring, the electric field strength vector would be directed along the axis. Alternatively, choose another small arc element lying diametrically opposite to the first element and draw their fields at point P to observe that their resultant field vector comes parallel to the axis. Same is the case for any small charge element on the ring and therefore it is concluded that the resultant field at any point on the axis points along the axis. Therefore the field at point P can be found as follows.

Hence for points l >> r, the ring charge distribution behaves like a point charge.

Field at Centre of the Ring

To get the electric field strength value at centre of the ring, we should simply substitute l=0 in the above obtained general expression of electric field. This gives zero field at the centre which can also be concluded from symmetry and uniformity of the charge distribution.

Maximum Possible Field Strength
Thus the magnitude of electric field intensity vector is maximum on both sides of the ring at a distance l = r/√2 from the ring's centre. To get that maximum value we should substitute this value of in the expression of E(l).

Approximate Plot of Field E(l)

The approximate plot of E(l) is shown below. 
The approximate plot has been drawn keeping in view the following three major points :
  1. The field value at the centre of the ring is zero. 
  2. The field value is maximum at l = r/√2 on both sides of the ring. 
  3. As tends to infinity the field value approaches to zero.
All of these three points can easily be concluded from the general expression of E(l). 

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