Question

2. The electric field due to a short dipole at a distance r on the axial line from its midpoint is the same as the electric field at a distance r’ on the equatorial line from its midpoint. Determine the ratio r: r’

Answer 1

Given:

The electric field due to a short dipole at a distance r on the axial line from its midpoint is the same as the electric field at a distance r’ on the equatorial line from its midpoint.

To find:

Ratio of r : r'

Calculation:

For convenience , r' is represented as r2.

At axial position:

$\therefore \: E = \dfrac{2kp}{ {r}^{3} }$

At equatorial position:

$\therefore \: E = \dfrac{kp}{ {(r2)}^{3} }$

Since the Field Intensity at both positions are same , we can say:

$\therefore \: E = \dfrac{2kp}{ {r}^{3} } = \dfrac{kp}{ {(r2)}^{3} }$

$= > \: \dfrac{ {r}^{3} }{ {(r2)}^{3} } = 2$

$= > \: {( \dfrac{r}{r2} )}^{3} = 2$

$= > \: \dfrac{r}{r2} = \sqrt[3]{2}$

Hence the required ratio is :

$\boxed{ \sf{\: r : r2 = \sqrt[3]{2} : 1}}$