Question

The electric potential at a point on the axis of an electric dipole at a distance r from its center is proportional to:

• A. \( \frac{1}{r} \)
• B. \( \frac{1}{r^2} \)
• C. \( \frac{1}{r^3} \)
• D. \( r^2 \)

Hint:

Dipole relations: \[ E \propto \frac{1}{r^3}, \quad V \propto \frac{1}{r^2} \]

Answer 1

The Correct Option is B

Step 1: Understand an electric dipole.
A dipole is a pair of equal and opposite charges set a small distance apart. We want how its electric potential changes as we move along its axis.

Step 2: Write the general potential.
The potential of a dipole at distance $r$ and angle $\theta$ from its axis is \[ V = \frac{1}{4\pi\varepsilon_0}\,\frac{p\cos\theta}{r^2}, \] where $p$ is the dipole moment.

Step 3: Use the axial condition.
A point on the axis means the angle is zero, so $\cos\theta = 1$. The formula becomes \[ V = \frac{1}{4\pi\varepsilon_0}\,\frac{p}{r^2}. \]

Step 4: Spot the fixed quantities.
The dipole moment $p$ and all the constants do not change as we move outward. Only $r$ changes.

Step 5: Read the proportionality.
Since everything else is fixed, the potential varies as one over $r$ squared. \[ V \propto \frac{1}{r^2}. \]

Step 6: State the answer.
The axial potential of a dipole falls off as the inverse square of the distance. \[ \boxed{\dfrac{1}{r^2}} \]