Question

show that workdone by conservative force (electric field) over a close look ​

Answer 1

Answer:

It depends on how you define conservative force field. If you define as a field such that work done along any closed path is 0, the question does not arrise, but then comes the question why does curl of such a field become zero. For the sake of this question let us assume that a conservative force field is a field such that the curl is 0 everywhere.  

Work done along a closed path is:

[math]W = \int\vec{F}(\vec{r}) \cdot d\vec{r}[/math]

If,

[math]\vec{\bigtriangledown} X  \vec{F}(\vec{r}) = \vec{0}[/math]

we can find at least one scalar function (in fact infinitely many separated by a constant, but any one suffices here), [math]V(\vec{r}) [/math], such that  

[math]\vec{F}(\vec{r}) = \vec{\bigtriangledown} V(\vec{r}) [/math] because,

[math ] \vec{\bigtriangledown}  X  \vec{\bigtriangledown} V(\vec{r}) = 0[/math], curl of a gradient vanishes

Hence, work done is nothing but:

[math]W = \int  \vec{\bigtriangledown} V(\vec{r})  \cdot d\vec{r}[/math] or

[math]W = V_{final} - V_{initial} [/math]  which is 0 for a closed path since initial and final positions are same.

 

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