Two points P1 and P2 are oppositely placed at same distance from centre of short dipole along the axial line of the dipole. If V1 and V2 are the electric potentials at theses points, then,

The relation between V1 and V2 is?

To understand the relationship between the electric potentials \( V_1 \) and \( V_2 \) at points \( P_1 \) and \( P_2 \) located along the axial line of a short dipole, we need to consider the nature of electric dipoles and how electric potential behaves in their vicinity.

Understanding Electric Dipoles

A short dipole consists of two equal and opposite charges, +q and -q, separated by a small distance, \( 2a \). The dipole moment \( \mathbf{p} \) is defined as \( \mathbf{p} = q \cdot 2a \) and points from the negative charge to the positive charge.

Electric Potential Due to a Dipole

The electric potential \( V \) at a point in space due to a dipole can be expressed as:

\( V = \frac{1}{4\pi \epsilon_0} \cdot \frac{\mathbf{p} \cdot \hat{r}}{r^2} \)

Here, \( \epsilon_0 \) is the permittivity of free space, \( \hat{r} \) is the unit vector in the direction from the dipole to the point where the potential is being calculated, and \( r \) is the distance from the dipole to that point.

Analyzing Points P1 and P2

In our scenario, points \( P_1 \) and \( P_2 \) are located at equal distances \( r \) from the center of the dipole, but on opposite sides. Therefore, we can denote their positions relative to the dipole moment:

• For point \( P_1 \): The potential \( V_1 \) is given by \( V_1 = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{r^2} \)
• For point \( P_2 \): The potential \( V_2 \) is given by \( V_2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{-p}{r^2} \)

Establishing the Relationship

From the expressions for \( V_1 \) and \( V_2 \), we can see that:

\( V_1 = -V_2 \)

This indicates that the electric potential at point \( P_1 \) is equal in magnitude but opposite in sign to the potential at point \( P_2 \). This is a direct consequence of the dipole's symmetry and the nature of electric fields generated by dipoles.

Conclusion

In summary, for two points \( P_1 \) and \( P_2 \) located along the axial line of a short dipole and equidistant from its center, the relationship between their electric potentials is:

V1 = -V2

This relationship highlights the fundamental properties of electric dipoles and their influence on the surrounding electric field.