Question

A circular ring of radius R with a total charge 2Q uniformly distributed along its circumference lies in the x y plane with its center at the origin.

(a) Find the electric field at a point with coordinates (0, 0, z0). Show all steps in your calculation. Don’t forget to represent the field in vector form - magnitude and direction!

(b) Find the locations along the z axis where the electric field has its largest values (don’t forget that because of the symmetry of the situation, there are two points where the field has it’s largest magnitude).

Answer 1

The linear charge density of the ring for the given problem can be determined as

  

So, total charge on the elemental length ' dl ' is

So, electric field due to this charge dQ at the point P on the Z axis at a distance z is given by

  

Now from the diagram, it is clear that the vertical component of the field will be cancelled out . So, the net electric field at point P is

  

Where,

So,

  

And hence the total electric field at P can be calculated as

(As i have taken two line elements on the ring, so i have choosen the limit from 0 to half circumference of the ring.)

or,

or,   

or,

This is the required value of electric field due to a uniformly charged rod on its axis.

(b) For electric field to be maximum

or,   

or,

or,

or,

or,

or,

For these two values of z , the electric field will be maximum.

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