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Hi I'm learning about Electric Potential, and the Work Done for moving a charge in an electric field. The last equation really does not make sense to me. Please see the image attached.

equation

I understand that to move a charge from point A to point B in an electric field (lets assume A is at a point which has a higher electric potential than point B) has a negative value for work done, and I understand the contents of the integral (simply the force required to move it multiplied by the length it moves). If the Electric Field was the other way around, the value for work done would be positive (it takes energy to move the charge from A to B now).

So why is the integral on the last line equal to zero? I'm probably missing something obvious here. Thanks

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This is simply an inaccurate statement by the author. The integral from $A$ to $B$ is not zero (unless the potential at both points is the same).

What the author meant is that work depends only on the potential at the respective points. This means that the integral along any closed curve would be zero.

So instead, they should have written something like:

$\displaystyle \oint_{C} q \vec{E} \cdot \mathrm{d} \vec{l} = 0$

Which is true for any closed curve $C$.

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