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GATE 2020 ME (ME1) – Question 37

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 2 marks · Multiple choice

Let $\vec f(x,y,z)=(x\hat i+y\hat j+z\hat k)/(x^2+y^2+z^2)^{3/2}$, with unit vectors of a right-handed rectangular Cartesian system. Find $\iint\vec f\cdot d\vec S$ over the inner and outer surfaces of the spherical shell formed by concentric spheres centred at the origin with radii 1 and 2.

  1. 0
  2. $2\pi$
  3. $4\pi$
  4. $8\pi$

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Correct answer: (A) 0

Explanation

The field is radial with magnitude 1/r². Outer-surface flux is (1/4)(4π·4)=4π. The inner boundary’s outward normal points toward the origin, giving −4π. Their sum is **0, A**. The origin is outside the shell, so divergence is zero throughout the material volume.