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Electric Potential Definition

Electric potential (V) is the electric potential energy per unit charge at a point. It is defined as the work done per unit charge to move a test charge from a reference point to that point.
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The statement of the theorem

In the context of a conservative force field F⃗=qE⃗\vec{F} = q\vec{E}, where qq is the test charge, the electric potential ϕ(r⃗)\phi(\vec{r}) at a point r⃗\vec{r} is defined as the negative line integral of the electric field E⃗\vec{E} along any differentiable path CC connecting a reference point r⃗0\vec{r}_0 to r⃗\vec{r}. Mathematically, this is expressed as:\n\nϕ(r⃗)=−∫CE⃗⋅dl⃗=−∫r⃗0r⃗E⃗⋅dl⃗\phi(\vec{r}) = -\int_{C} \vec{E} \cdot d\vec{l} = -\int_{\vec{r}_0}^{\vec{r}} \vec{E} \cdot d\vec{l} \n\nSince E⃗\vec{E} is conservative, the integral is path-independent. Furthermore, the electric field E⃗\vec{E} is related to the potential ϕ\phi via the negative gradient operator ∇\nabla: \n\nE⃗(r⃗)=−∇ϕ(r⃗)\vec{E}(\vec{r}) = -\nabla\phi(\vec{r})\n\nThis relationship implies that the potential ϕ\phi must satisfy Laplace's equation in charge-free regions (ρ=0\rho=0):\n\n∇2ϕ=0\nabla^2\phi = 0