Q. 634.3( 4 Votes )

Consider a circular ring of radius r, uniformly charged with linear charge density λ. Find the electric potential at a point on the axis at a distance x from the center of the ring. Using this expression for the potential, find the electric field at this point.

Answer :


Given:
Radius of the circular ring: r
Linear charge density : λ
Distance of a point from the center of the ring : x

From the diagram we can see that,
Point P is at a distance x from the center of the ring.
Point P is at a distance of
from the surface of the ring: r’ =
Circumference of the ring is : L = 2π r
Formula used:
We can see that, Electric field as p is resolved into vertical and horizontal components. As the ring is symmetric, vertical components are cancelled out and horizontal components add.
Thus Enet =Ecosθ , where θ is the angle between x and
.
We know that,
Where, λ is the linear charge density, Q is the Total charge due to whole ring and L is the circumference of the ring.
Potential at a point due to charge Q is:

k is a constant and k= =9× 109 Nm2C-2 . q is the point charge.
If we substitute value of k then:


Hence, Electric potential at a point x from the center of the ring is
.
Net electric field at P is Ecosθ and

From the figure:





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