Question

Determine the divergence of F = 30 i + 2xy j + 5xz^2 k at (1,1,-0.2) and state the nature of the field.

(a) 1, solenoidal

(b) 0, solenoidal

(c) 1, divergent

(d) 0, divergent

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I need to ask this question from Divergence topic in section Vector Calculus of Electromagnetic Theory

Answer 1

Right option is (b) 0, solenoidal

The best explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz^2) = 0 + 2x + 10xz = 2x + 10xz

Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.

Alternate/Shortcut: Without calculation, we can easily choose option “0, solenoidal”, as by theory when the divergence is zero, the vector is solenoidal. “0, solenoidal” is the only one which is satisfying this condition.