Consider I= V2(1/r)\,dV where r=x2+y2+z2. Which statement(s) is/are correct?

The integrand diverges as $r\to\infty$

GATE 2024 PH – Question 30

Consider $I=\int_V\nabla^2(1/r)\,dV$ where $r=\sqrt{x^2+y^2+z^2}$. Which statement(s) is/are correct?

Show answer and explanation

Correct answer: (A) $I=-4\pi$ if the origin is inside V; (B) The integrand vanishes for $r\ne0$; (C) $I=0$ if the origin is outside V

The function $\dfrac1r$ is the potential of a point charge, and its Laplacian is
$$\nabla^2\frac1r=-4\pi\,\delta^3(\mathbf r).$$

**A. $I=-4\pi$ if the origin is inside V.** True: the integral of the delta function over a volume containing the origin is 1. ✓
**B. The integrand vanishes for $r\neq0$.** True: away from the origin $\dfrac1r$ is harmonic. ✓
**C. $I=0$ if the origin is outside V.** True: the integrand is zero everywhere inside V. ✓
**D. The integrand diverges as $r\to\infty$.** False: it is zero for large $r$.

Answer **A, B and C**.