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2015-02-22T22:33:41+05:30

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Work done in an electric field by transporting an object having an electric charge q coulombs from a point having a potential V1 volts to a point having potential V2 is given by
               W = q (V2 - V1) 

    The work done is also defined as the force * displacement. 
W= \int\limits^{P_2}_{P_1} {F} \, ds\\\\=\int\limits^{P_2}_{P_1} {[q*E(x,y)]} \, ds\\\\=q*\int\limits^{P_2}_{P_1} {E(x,y)} \, ds\\\\=q*[V]_{V_1}^{V_2}=q*[V_2-V_1]\\\\as,\ E(x,y)=\frac{dV}{ds}\\Electric\ field\ at\ a\ point\ (x,y)=space\ gradient\ of\ electric\ potential

So no matter whichever path S is taken from point P1 to point P2, the integral will have only one value.

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