Work done to move a charge from A to B is independent of the path taken. But how does this allow us to define the potential energy of a charge and electric potential at every point in space?
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$\begingroup$ You chose a common reference, say, the infinity. $\endgroup$– stafusaCommented Sep 18, 2017 at 11:53
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This is a result of the converse gradient theorem: suppose $\int_{\gamma}\vec{E}.d\vec{\sigma}$ is independent of the chosen path $\gamma$ between two points $A$ en $B$. Then $\exists V: \vec{E}=-\nabla V$. Here V is a scalar field wich assigns a numerical value to every point in space: the electric potential. It is merely a convention to choose $V$ so that points at infinity have potential $0$.
