The main objective of this article is to calculate the Electric Field due to a Uniformly Charged Ring at a Point on its Axis. Prerequisite of this calculation is the knowledge of Electric Field due to Point Charge and Line Charge Distribution. We recommend to visit those article before continuing. Moreover, you can browse another theoretically similar article on Gravitational Field due to a Uniform Ring at a Point on its Axis for a comparative study.
Electric Field due to Uniformly Charged Ring at a Point on its Axis
Here we will calculate the electric field at an axial point P due to a uniformly charged ring of charge +Q. The center of the ring is at point O. From the figure shown below, the circumference of the ring makes an angle with the line OP drawn from the center of the ring to the test point P. We will consider a very small charge element (shown as red charge element), on the circumference of the ring. The center of this charge element is at point A. Now to calculate the field at point P due to this charge element, we will use the expression of the electric field derived in our previous article, Electric Field due to Point Charge.
The direction of this field is along . Now, we can resolve this field vector into two components, one along the positive x-direction and another along the negative y-direction. This is shown as the red vectors in the diagram above. The x component of the electric field due to the charge element is
The y component of the electric field due to the charge element is
To calculate the total electric field at point P due to this circular charged ring, we have to integrate over the whole ring. However, due to the axial symmetry of the total system, the y component will vanish and we will have the x component only. What does it mean? In order to explain this, we have considered another charge element (shown in green). Electric field due to this charge element is shown as a green vector. Horizontal and vertical components are also shown in green. Now, for each charge there will be another positioned at a radially opposite point on the circumference of the charged ring. The y component of the electric field due to these two charges cancel each other, whereas the x components add up. This is true for each and every point on the charged circular ring. Thus we are left with x component of the field only.
The direction of the field along . This is the final expression for the electric field due to uniformly charged ring at a point on its axis. In the next article, we will use this expression to calculate the Electric Field due to a Uniformly Charged Disc. Can you solve the problem for an off-axis point?