Module 2 : Electrostatics
Lecture 8 : Electrostatic Potential
  where $\vec x_0$is a reference point such that $\phi(\vec x_0)=0$.
  In one dimension,
 
\begin{displaymath}\phi( x) = \int_{ x_0}^{ x} E(x)dx\end{displaymath}
  Differentiate both sides with respect to the upper limit of integration, i.e. $x$
\begin{displaymath}\frac{\partial\phi}{\partial x} = -E(x)\end{displaymath}
  In three dimension, we use the fundamental theorem on gradients
 
\begin{displaymath}\phi(\vec x)-\phi(\vec x_0) = \int_{\vec x_0}^{\vec x} (\vec\nabla\phi) \cdot d\vec l\end{displaymath}
  which gives
 
\begin{displaymath}\phi(\vec x) = \int_{\vec x_0}^{\vec x} (\vec\nabla\phi) \cdot d\vec l\end{displaymath}
   
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