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Electric Field Definition

The electric field (E) is the force per unit charge experienced by a test charge at a point, defined as E = F/q.
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The statement of the theorem

The electric field E⃗(r⃗)\vec{E}(\vec{r}) at a point r⃗\vec{r} in a region of space containing a charge distribution ρ(r⃗′)\rho(\vec{r}') is formally defined as the limit of the force F⃗\vec{F} exerted by the source charges on an infinitesimally small test charge q0q_0 placed at r⃗\vec{r}, normalized by q0q_0. Mathematically, this is expressed as:\n\nE⃗(r⃗)=lim⁡q0→0F⃗source→q0q0\vec{E}(\vec{r}) = \lim_{q_0 \to 0} \frac{\vec{F}_{source \to q_0}}{q_0} \n\nIn the context of a conservative field derived from a scalar potential ϕ(r⃗)\phi(\vec{r}), the electric field is rigorously defined as the negative gradient of the potential energy per unit charge:\n\nE⃗(r⃗)=−∇ϕ(r⃗)\vec{E}(\vec{r}) = -\nabla \phi(\vec{r})\n\nWhere ϕ(r⃗)\phi(\vec{r}) is the electrostatic potential, which itself is defined by the volume integral over the charge density ρ(r⃗′)\rho(\vec{r}'):\n\nϕ(r⃗)=14πϵ0∫ρ(r⃗′)∣r⃗−r⃗′∣d3r′\phi(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}')}{\left|\vec{r} - \vec{r}'\right|} d^3r'