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Question #42493

Difficulty: 8/10 | 🟢 Free

The Question

Four identical point charges, each of magnitude $+Q$ , are fixed at the coordinates $(a, a, 0)$ , $(-a, a, 0)$ , $(-a, -a, 0)$ , and $(a, -a, 0)$ in vacuum. A small test charge $+q$ is placed at the origin $(0,0,0)$ . Let $U(x,y,z)$ be the electrostatic potential energy of the test charge. Which of the following represents the values of the second partial derivatives $\left( \frac{\partial^2 U}{\partial x^2}, \frac{\partial^2 U}{\partial z^2} \right)$ evaluated at the origin? (Here $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$\left( \frac{kqQ}{\sqrt{2}a^3}, -\frac{\sqrt{2}kqQ}{a^3} \right)$
B.
$\left( \frac{\sqrt{2}kqQ}{a^3}, -\frac{\sqrt{2}kqQ}{a^3} \right)$
C.
$\left( -\frac{kqQ}{\sqrt{2}a^3}, \frac{\sqrt{2}kqQ}{a^3} \right)$
D.
$\left( \frac{kqQ}{\sqrt{2}a^3}, \frac{kqQ}{\sqrt{2}a^3} \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Equilibrium Stability|Superposition of Potentials|Potential Gradient Direction|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Earnshaw's Theorem and Laplace's equation ( $\nabla^2 V = 0$ ) to determine the stability of electrostatic equilibrium.

Students often fall into the trap of assuming that a point with zero electric field implies stable equilibrium in all directions, or they attempt to manually calculate the complex second derivative in the $xy$ -plane.

By requiring the calculation of the Hessian matrix components, the problem forces students to recognize that the trace of the Hessian must be zero in a vacuum.

Thus, a restoring force in the $xy$ -plane (positive curvature) mathematically necessitates an expelling force along the $z$ -axis (negative curvature), proving the equilibrium is a saddle point.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Equilibrium Stability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free