Plotting E versus r for a Dipole on the Axial Line

The electric field due to a dipole of length 2a at a point r on the axial line is given by the equation:

E = (p/4πε0) [(2r^3 - 3ar^2)/(r^2 + a^2)^(5/2)]

where p is the magnitude of the electric dipole moment, ε0 is the permittivity of free space, and a is the distance between the charges that make up the dipole.

To plot E versus r for r>>a, we can make the following approximations:

- For r>>a, we can neglect the term a^2 in the denominator of the equation, since it will be much smaller than r^2.
- We can also neglect the term 2r^3 in the numerator, since it will be much larger than 3ar^2 for large values of r.

With these approximations, the equation for the electric field simplifies to:

E = (p/4πε0) (3cosθ/r^3)

where θ is the angle between the dipole moment and the axial line.

Plotting this equation for various values of θ, we get a graph that looks like:

![Graph of E versus r for a dipole on the axial line](

https://www.edurev.in/api/listing/getimage?imgid=5f9e4c6e8da5f) https://www.edurev.in/api/listing/getimage?imgid=5f9e4c6e8da5f)

As we can see from the graph, the electric field decreases rapidly with distance from the dipole, and the shape of the curve depends on the angle θ. When θ=0 (i.e., the dipole moment is aligned with the axial line), the curve is symmetric around the origin and has a maximum at r=0. When θ=90° (i.e., the dipole moment is perpendicular to the axial line), the curve has a minimum at r=0 and approaches zero more slowly as r increases.