Question

The work done by the force $\vec{F}-x^{2} \hat{i} 1 y^{2} \hat{j}$ around the path shown in the figure is: (a) $\frac{2}{3} a^{3}$ (b) zero (c) $a^{3}$ (d) $\frac{4}{3} a^{3}$

   The work done by the force $\vec{F}-x^{2} \hat{i} 1 y^{2} \hat{j}$ around the path shown in the figure is:
(a) $\frac{2}{3} a^{3}$
(b) zero
(c) $a^{3}$
(d) $\frac{4}{3} a^{3}$

Practice Problem in Physics for the JEE Main and Advanced

Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar 2015 Edition

Chapter 5, Problem 31

Instant Answer

Solved by Expert Prabhu Ramji

01/20/2022

Step 1

In this case, the force is given by $\vec{F} = -x^{2} \hat{i} + y^{2} \hat{j}$ and the path is a square with side length $a$. Show more…

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The work done by the force $\vec{F}-x^{2} \hat{i} 1 y^{2} \hat{j}$ around the path shown in the figure is: (a) $\frac{2}{3} a^{3}$ (b) zero (c) $a^{3}$ (d) $\frac{4}{3} a^{3}$

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Key Concepts

Line Integrals in Vector Fields

The line integral in the context of force fields represents the work performed by a force when moving an object along a curve. It is computed by taking the dot product of the force vector and the infinitesimal displacement vector and integrating this product along the path. This integral is crucial in determining the total work done, especially in scenarios involving closed loops.

Conservative Vector Fields

A vector field is conservative if the work done along any path between two points is independent of the path taken. This is typically verified by checking if the curl of the field is zero in a simply connected region, which implies the existence of a scalar potential function. For conservative fields, the work done around any closed path is zero.

Curl and its Role in Field Characterization

The curl of a vector field quantifies the field's tendency to induce rotation around a point. It is an essential tool in determining whether a vector field is conservative. A zero curl in a simply connected domain indicates that the field is conservative, meaning that the work done over any closed curve in that field is zero.

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