Electrostatic Potential due to a Ring of Charge at a Point on its Axis

Electrostatic Potential due to a Ring of Charge at a Point on its Axis

In the main Electrostatic Potential article, we have rigorously defined the electric potential and have seen the expressions of electric potential due to different symmetric situations. Here we will derive the expression for the electrostatic potential due to a ring of charge at a point on its axis. The main idea behind the calculation of any electrostatic or electric potential is the calculation of work done to move a unit positive charge from a point of reference, generally chosen to be infinity to the field of influence. However here, we will skip that methodology and will calculate the potential due to a small differential charge element and will integrate over the ring to get the total potential at a point on its axis. No surprise that this methodology is very similar to the calculation of Gravitational Potential due to a Uniform Ring mass system. Also, notice that this is an example of Electrostatic Potential due to a Uniform Line Charge distribution.

Electrostatic Potential due to a Ring of Charge at a Point on its Axis

Here we will calculate the electrostatic potential due to a ring of charge of radius R at an axial point P. The uniformly charged ring is centered at O. The distance of P from the center O is Z. From the diagram shown below, the circumference of the ring makes an angle \theta with the line OP drawn from the center of the ring. We will consider a differential charge element dq. Now the electrostatic potential at point P due to this differential charge element dq is.

electrostatic potential due to a ring charge distribution

    \[ dV(r)=\frac{1}{4\pi\epsilon_0}\frac{dq}{r} \]

Thus

    \[ V(r)=\frac{1}{4\pi\epsilon_0 r}\int_{ring}dq \]

    \[ V(r)=\frac{1}{4\pi\epsilon_0 r}Q \]

Where Q is the total charge of the ring and is equal to 2\pi\lambda R. Here \lambda is the line charge density of the ring. Also note that |\vec{r}| is independent of the position of the differential charge element and therefore comes out of the integration.

    \[ V(r)=\frac{\lambda R}{2\epsilon_0 r} \]

    \[ \boxed{V(r)=\frac{\lambda R}{2\epsilon_0 \sqrt{R^2+Z^2}}} \]

This is the final expression for the potential due to a ring of charge at a point on its axis. The electrostatic potential is a scalar quantity which makes this calculation very easy.

At a very large distance i.e. Z>>R, the approximate potential is

    \[ V(r)=\frac{\lambda R}{2\epsilon_0 Z} \]

In the next article, we will use this expression to calculate the Electrostatic Potential due to a Charged Disc.

Help us to improve. Rate this article. It matters.
1 Star2 Stars3 Stars4 Stars5 Stars (7 votes, average: 4.86 out of 5)

Loading...