GATE EC 2024 Set 1 — Question 54
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Question
Correct answer
(A) For every closed path, we have _ (F₁ dx + F₂ dy + F₃ dz) = 0.; (B) There exists a differentiable scalar function f(x, y, z) such that F₁ = (∂ f)/(∂ x), F₂ = (∂ f)/(∂ y), F₃ = (∂ f)/(∂ z).; (D) (∂ F₃)/(∂ y) = (∂ F₂)/(∂ z), (∂ F₁)/(∂ z) = (∂ F₃)/(∂ x), (∂ F₂)/(∂ x) = (∂ F₁)/(∂ y)
Solution
(A) True: For a conservative field, the line integral around any closed loop is zero.
(B) True: A conservative field can always be expressed as the gradient of a scalar potential function , i.e., .
(D) True: A conservative field is irrotational, meaning its curl is zero (). The components of the curl being zero lead exactly to the equalities in option (D).
(C) False: This is the condition for a solenoidal (divergence-free) field, which is not a requirement for a field to be conservative.
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