Question

Electric potential due to a non uniformly charged ring having total charge q and radius r at centre of ring is

Answer 1

Given:

A non uniformly charged ring having total charge q and radius r has been provided.

To find:

Electric potential at the centre of ring.

Calculation:

Let's consider that a small elemental portion of ring contributes to dV (potential) of ring. That small part of ring consists of dq charge.

$\sf{ \therefore \: potential \: due \: to \: elemental \: part =dV }$

$\sf{ = > \: dV = \dfrac{k(dq)}{r} }$

Integrating on both sides :

$= > \displaystyle \int \sf{ \: dV = \int \: \dfrac{k(dq)}{r} }$

$= > \displaystyle \sf{\:V = \int \: \dfrac{k(dq)}{r} }$

$= > \displaystyle \sf{\:V = \: \dfrac{k}{r} \: \int(dq) }$

$= >\sf{\:V = \: \dfrac{k}{r} \times q }$

$= >\sf{\:V = \: \dfrac{kq}{r} }$

Putting value of Coulomb's Constant :

$= >\sf{\:V = \: \dfrac{q}{4\pi\epsilon_{0}r} }$

So , final answer is :

$\boxed{ \red{\sf{\:V = \: \dfrac{q}{4\pi\epsilon_{0}r} }}}$