GATE 2022 EE Q.45: solution and answer

GATE 2022 · EEQ.45 · 2 marks · MCQ

If the magnetic field intensity $(H)$ in a conducting region is given by the expression, $H = x^{2} \hat{i} + x^{2} y^{2} \hat{j} + x^{2} y^{2} z^{2} \hat{k}$ A/m. The magnitude of the current density, in A/m $^{2}$ , at $x = 1$ m, $y = 2$ m, and $z = 1$ m, is

(A)
8
(B)
12
(C)
16
(D)
20

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Answer and solution

ANSWER

The magnitude of the current density at (1, 2, 1) is 12 A/m², so the answer is option (B).

Official answer: BSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

Ampere's law in point form. The field is static, so there is no displacement current and $J = \nabla \times H$ . The components are $H_{x} = x^{2}$ , $H_{y} = x^{2} y^{2}$ and $H_{z} = x^{2} y^{2} z^{2}$ .
$x$ -component: $J_{x} = \frac{\partial H _{z}}{\partial y} - \frac{\partial H _{y}}{\partial z} = 2 x^{2} y z^{2} - 0 = 2 x^{2} y z^{2}$
$y$ -component: $J_{y} = \frac{\partial H _{x}}{\partial z} - \frac{\partial H _{z}}{\partial x} = 0 - 2 x y^{2} z^{2} = - 2 x y^{2} z^{2}$
$z$ -component: $J_{z} = \frac{\partial H _{y}}{\partial x} - \frac{\partial H _{x}}{\partial y} = 2 x y^{2} - 0 = 2 x y^{2}$
Evaluate at $(1, 2, 1)$ : $J_{x} = 2 (1)^{2} (2) (1)^{2} = 4, J_{y} = - 2 (1) (2)^{2} (1)^{2} = - 8, J_{z} = 2 (1) (2)^{2} = 8$ This gives $J = 4 \hat{i} - 8 \hat{j} + 8 \hat{k} A/m^{2}$ .
Magnitude: $∣ J ∣ = 4^{2} + (- 8)^{2} + 8^{2} = 16 + 64 + 64 = 144 = 12 A/m^{2}$ This is option (B).

KEY IDEA

For a static field, the point form of Ampere's circuital law is $J = \nabla \times H$ . Take the curl symbolically first, substitute the point only after that, and then find the magnitude from all three components.

WATCH OUT FOR

Substituting the point into $H$ before differentiating, which makes every derivative zero. Another mistake is getting a curl component's sign wrong: $J_{y} = \partial H_{x} / \partial z - \partial H_{z} / \partial x$ , not the reverse. Adding the components ( $4 + 8 + 8 = 20$ , option D) instead of taking the root of the sum of their squares, or dropping $J_{x}$ ( $128 \approx 11.3$ ), also gives a wrong answer.

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GATE 2022 EE Q.45: solution and answer

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