Question

What will be the ratio of the electric field at a point in the axis and an equidistant point on the equatorial line of a dipole?

Answer 1

Hint: An electric dipole is a system of two charges of equal magnitude but with opposite charges separated by a distance. In order to find the electric field at these points, one has to be well versed in where they are. Some people have trouble determining axial and equatorial lines.

Formulas used:
The formula to find the value of electric field of a point in the axial line is,
$$E_{axial}={\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{2p}{r^3}}$$
The formula to find the value of electric field of a point in the equatorial line is, $$E_{equatorial}={\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{p}{r^3}}$$

Complete step by step answer:
The formula to find the value of electric field of a point in the axial line is,
$$E_{axial}={\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{2p}{r^3}}$$
The formula to find the value of electric field of a point in the equatorial line is, $$E_{equatorial}={\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{p}{r^3}}$$
The only value that can be a variable is the distance, $$r$$ . Taking the condition given in the question, we know that the points are equidistant which means that they are distanced equally. Hence, we can take one same value for the value of $$r$$ .

Since we have the values of the electric fields at the respective points, now we find the ratio,
$${\displaystyle \frac{E_{equatorial}}{E_{axial}}}={\displaystyle \frac{{\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{p}{r^3}}}{{\displaystyle \frac{1}{4\pi {\mathrm{E}}_0}}{\displaystyle \frac{2p}{r^3}}}}$$
Cancelling out the constants, we arrive at the ratio.
$$\therefore {\displaystyle \frac{E_{equatorial}}{E_{axial}}}={\displaystyle \frac{1}{2}}$$
This means that the value of electric field on any point on the axial line will be twice as much as the value of electric field in the equatorial line provided that they are equidistant.

Therefore, the ratio will be $$1:2$$.

Note: Electric field is the physical field possessed by charged particles exerting a force on all other particles that are charged in their field. It can either be attracting or repelling. Ratio is the quantitative physical relation between two amounts of the same quantity showing how many times one amount is contained within another. Since ratio is a division of amounts of the same quantity, it is a dimensional quantity.