Two electric dipoles, $A, B$ with respective dipole moments $\vec{d}_A = - 4 \,qa \,\hat{i}$ and $\vec{d}_B = - 2 \,qa\, \hat{i}$ placed on the x-axis with a separation $R$, as shown in the figure The distance from $A$ at which both of them produce the same potential is :

Question

The electrostatic potential due to an electric dipole at a distance $ r $ varies as:

Answer 1

To determine the distance from dipole $ A $ at which both dipoles $ A $ and $ B $ produce the same potential, we need to use the formula for the electric potential $ V $ due to a dipole at a point along its axial line:

V = \frac{1}{4\pi \varepsilon_0} \cdot \frac{\vec{d} \cdot \hat{r}}{r^2}

Since dipoles $ A $ and $ B $ are aligned along the x-axis with dipole moments $ \vec{d}_A = - 4 \,qa \,\hat{i}$ and $ \vec{d}_B = - 2 \,qa\, \hat{i} $, consider a point $ P $ at a distance $ x $ from dipole $ A $ where x \lt R (as both dipoles produce equal potential at that point).

Step 1: Calculate the potential at point $ P $ due to dipole $ A $

Electric potential at distance $ x $ due to dipole $ A $:

V_A = \frac{-4 \,qa}{4\pi \varepsilon_0 \, x^2}

Step 2: Calculate the potential at point $ P $ due to dipole $ B $

The distance from dipole $ B $ is $ R - x $.

Electric potential at distance $ R - x $ due to dipole $ B $:

V_B = \frac{-2 \,qa}{4\pi \varepsilon_0 \, (R - x)^2}

Step 3: Set the potentials equal

For the potential due to both dipoles to be equal at point $ P $, equate $ V_A $ and $ V_B $:

\frac{-4 \,qa}{x^2} = \frac{-2 \,qa}{(R - x)^2}

Canceling the common terms and simplifying:

\frac{4}{x^2} = \frac{2}{(R-x)^2}

\frac{2}{x^2} = \frac{1}{(R-x)^2}

Taking reciprocal and square roots gives:

x = \frac{R-x}{\sqrt{2}}

Solving for $ x $:

\sqrt{2}x = R - x

x(\sqrt{2} + 1) = R

x = \frac{R}{\sqrt{2} + 1}

We apply the property of surds for simplicity:

x = \frac{R}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1}

x = \frac{R(\sqrt{2} - 1)}{1}

x = \frac{\sqrt{2}R}{\sqrt{2} - 1}

Therefore, the correct answer is:

\frac{\sqrt{2}R}{\sqrt{2} - 1}

Answer 2