The GATE Grind

GATE 2014 EC (EC4) – Question 65

Electromagnetics · Maxwell's Equations · 2 marks · Multiple choice

$\mathbf E=-(2y^3-3yz^2)\hat x-(6xy^2-3xz^2)\hat y+6xyz\hat z$ is described as the electric field in a source-free region. Find a valid electrostatic potential.

  1. $xy^3-yz^2$
  2. $2xy^3-xyz^2$
  3. $y^3+xyz^2$
  4. $2xy^3-3xyz^2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $2xy^3-3xyz^2$

Explanation

Since E=−∇ψ, integrating −Ex=2y³−3yz²with respect to x givesψ=2xy³−3xyz²+g(y,z). Differentiating this expression reproduces the supplied yand zcomponents when g is constant. Therefore **ψ=2xy³−3xyz²,D**, up to an additive constant. The source-free description is inconsistent:∇·E=−6xy, generally nonzero in a uniform-permittivity medium.