Assertion (A): For a force field given by F = ( 2 x y + z 2 ) i ^ + ( x 2 + 2 yz ) j ^ + ( 2 x z + y 2 ) k ^ , the work done by this force in moving a particle from ( 0 , 0 , 0 ) to ( 1 , 1 , 1 ) is

Question

Assertion (A): For a force field given by $F = (2 x y + z^{2}) \hat{i} + (x^{2} + 2 yz) \hat{j} + (2 x z + y^{2}) \hat{k}$ , the work done by this force in moving a particle from $(0, 0, 0)$ to $(1, 1, 1)$ is independent of the path taken.

Reason (R): A force is conservative if its curl is zero, i.e., $\nabla \times F = 0$ , and for such a force, work done between two points depends only on the initial and final positions.

TestHub · Accepted Answer

A force is conservative if its curl is zero. Let F x ​ = 2 x y + z 2 , F y ​ = x 2 + 2 yz , F z ​ = 2 x z + y 2 . We check the curl components: ∂ y ∂ F z ​ ​ − ∂ z ∂ F y ​ ​ = ( 2 y ) − ( 2 y ) = 0 . ∂ z ∂ F x ​ ​ − ∂ x ∂ F z ​ ​ = ( 2 z ) − ( 2 z ) = 0 . ∂ x ∂ F y ​ ​ − ∂ y ∂ F x ​ ​ = ( 2 x ) − ( 2 x ) = 0 . Since ∇ × F = 0 , the force is conservative, and thus the work done is path independent. Reason R correctly defines a conservative force and its property regarding work done.

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