Question:
Two point charges \(2q\) and \(8q\) are placed at a distance \(r\) apart. Where should a third charge \(-q\) be placed between them, so that the electrical potential energy of the system is minimum?
Two point charges \(2q\) and \(8q\) are placed at a distance \(r\) apart. Where should a third charge \(-q\) be placed between them, so that the electrical potential energy of the system is minimum?
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Potential energy is minimum when the third charge is placed at infinity, not between the charges.
Updated On: Apr 20, 2026
- At a distance of \(r/3\) from \(2q\)
- At a distance of \(2r/3\) from \(2q\)
- At a distance of \(r/16\) from \(2q\)
- None of the above
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
Total potential energy of three charges: \[ U = k\left[\frac{2q(8q)}{r} + \frac{(-q)(2q)}{x} + \frac{(-q)(8q)}{r-x}\right] \]
Step 2: Detailed Explanation:
Let charge \(-q\) be at distance \(x\) from \(2q\). Then total energy: \[ U = k\left[\frac{16q^2}{r} - \frac{2q^2}{x} - \frac{8q^2}{r-x}\right] \] For extrema: \[ \frac{dU}{dx} = 0 \Rightarrow \frac{2}{x^2} - \frac{8}{(r-x)^2} = 0 \] Solve: \[ \frac{x}{r-x} = \frac{1}{2} \Rightarrow x = \frac{r}{3} \] Check nature: at \(x = \frac{r}{3}\), second derivative is negative, so energy is maximum (not minimum). Hence, no minimum potential energy exists between the charges.
Step 3: Final Answer:
\[ \boxed{\text{None of the above}} \]
Total potential energy of three charges: \[ U = k\left[\frac{2q(8q)}{r} + \frac{(-q)(2q)}{x} + \frac{(-q)(8q)}{r-x}\right] \]
Step 2: Detailed Explanation:
Let charge \(-q\) be at distance \(x\) from \(2q\). Then total energy: \[ U = k\left[\frac{16q^2}{r} - \frac{2q^2}{x} - \frac{8q^2}{r-x}\right] \] For extrema: \[ \frac{dU}{dx} = 0 \Rightarrow \frac{2}{x^2} - \frac{8}{(r-x)^2} = 0 \] Solve: \[ \frac{x}{r-x} = \frac{1}{2} \Rightarrow x = \frac{r}{3} \] Check nature: at \(x = \frac{r}{3}\), second derivative is negative, so energy is maximum (not minimum). Hence, no minimum potential energy exists between the charges.
Step 3: Final Answer:
\[ \boxed{\text{None of the above}} \]
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