Normal Derivative

#Physics

$\displaystyle \frac{\partial V }{\partial n}=\nabla V\cdot \hat{n}$

$\displaystyle \sigma=-{\varepsilon}_{0}\frac{\partial V }{\partial n}$

• Surface charge

$\displaystyle \partial_{n}V_{+}|{x=0}-\partial{n}V_{-}|{x=0}=-\frac{\sigma}{{\varepsilon}{0}}$

• $\displaystyle \partial_{n}V_{+}$ is the normal derivative of the voltage just above a surface
• $\displaystyle \sigma$ is the surface charge density
• $\displaystyle {\varepsilon}_{0}$ is the permitivity of free space