Question

Two point charges \(-q\) and \(+q\) are located at points \((-a, 0, 0)\) and \((a, 0, 0)\) respectively. Find the electrostatic potential at point \((x, 0, 0)\) where \(x \gg a\).

Answer 1

Potential due to two charges

Step 1: General expression of potential
Electric potential at point \(x\) due to point charges: \[ V = \frac{kq_1}{r_1} + \frac{kq_2}{r_2} \] Here:
• Charge \(+q\) at \(x = a\)
• Charge \(-q\) at \(x = -a\) So: \[ V = \frac{kq}{x-a} - \frac{kq}{x+a} \]

Step 2: Simplify expression
\[ V = kq \left( \frac{1}{x-a} - \frac{1}{x+a} \right) \] Take LCM: \[ V = kq \cdot \frac{(x+a)-(x-a)}{(x-a)(x+a)} \] \[ V = kq \cdot \frac{2a}{x^2 - a^2} \]

Step 3: Apply condition \(x \gg a\)
If \(x \gg a\), then: \[ x^2 - a^2 \approx x^2 \] So: \[ V \approx \frac{2kqa}{x^2} \] Final Answer: \[ \boxed{V \approx \frac{2kqa}{x^2}} \]