In a region of space the force on a electron is $\vec{F} = Cx \hat{j},$ where C is a positive constant. The electron moves around a square loop in the xy-plane. Calculate the work done on the electron by the force $\vec{F}$ during a counterclockwise trip around the square. Is this force conservation or non conservative?

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${CL}^{2}, Conservative$

$\frac{{CL}^{2}}{2}, Non conservative$

${CL}^{2}, Non conservative$

$\frac{{CL}^{2}}{4}, Non conservative$

answer is C.

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Detailed Solution

IDENTIFY and SET UP: The force $\vec{F}$ is not constant, and in general it is not in the same direction as the displacement. To calculate the work done by $\vec{F}$ . we'll use the general expression for work, Eq.(6.14): $W = \int_{p_{1}}^{p_{2}} \vec{F} . \vec{d l}$

where $\vec{d l}$ is an infinitesimal displacement. We'll calculate the work done on each leg of the square separately, and add the result to find the work done on the round trip. If this round-trip work is zero, force $\vec{F}$ is conservative and can be represented by a potential energy function.

EXECUTE: On the first leg, from $(0, 0)$ to $(L, 0)$ , the force is every where perpendicular to the displacement. So $\vec{F} . \vec{d l} = 0,$ and the work done on the first leg is $W_{1} = 0$ .The force has the same value $\vec{F} = C L \hat{j}$ everywhere on the second leg, from $(L, 0)$ to $(L, L)$ . The displacement on this leg is in the $+ y$ -direction, so $\hat{d l} = d y \hat{j}$ and $\vec{F .} \vec{d l} = C L j . d y \hat{j} = C L d y$

The work done on the second leg is then $W_{2} = \int_{(L, 0)}^{(L, L)} \vec{F} . \vec{d l} = \int_{Y = 0}^{y = L} C L d y = C L \int_{0}^{L} d y = C L^{2}$

On third leg, from $(L, L)$ to $(0, L) . \vec{F}$ is again perpendicular to the displacement and so $W_{4} = 0 .$ The work done by $\vec{F}$ on the round trip is therefore $W = W_{1} + W_{2} + W_{3} + W_{4} = 0 + C L^{2} + 0 + 0 = C L^{2}$

The starting and ending points are the same but the total work done by $\vec{F}$ is not zero. This is a non conservative force; it cannot be represented by a potential-energy functions.

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