Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Relationship between Potential and Field

E = -∇V, where E is the electric field and V is the electric potential.
📜

The statement of the theorem

In the domain Ω⊂R3\Omega \subset \mathbb{R}^3, let V(r⃗)V(\vec{r}) be a scalar potential field, V:Ω→RV: \Omega \to \mathbb{R}, and let E⃗(r⃗)\vec{E}(\vec{r}) be the associated electric field vector field. The relationship is defined by the negative gradient operation, which quantifies the rate of change of the potential in the direction of steepest descent (the direction of the field). Formally, the electric field E⃗\vec{E} is the negative gradient of the electric potential VV: \n\nE⃗=−∇V\vec{E} = -\nabla V \n\nIn Cartesian coordinates, this relationship expands to:\n\nE⃗=−(∂V∂xi^+∂V∂yj^+∂V∂zk^)\vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \n\nThis implies that the field E⃗\vec{E} is a conservative vector field, satisfying the condition ∇×E⃗=0\nabla \times \vec{E} = 0, which is a direct consequence of VV being a scalar potential function.
Source: Wikipedia