ece106: add voltage intro

docs/1b/ece106.md

@@ -310,3 +310,22 @@ Generally:
 
 1. Determine $\vec E$ outside the slab.
 2. Set one outside surface and one inside surface as a pillbox and apply rules.
+
+## Electrostatic potential
+
+At a point $P$, the electrostatic potential $V_p$ or voltage is the work done per unit positive test charge from infinity to bring it to point $P$ by an external agent.
+
+$$
+V_p=\lim_{q\to 0^+}\frac{W_i}{q} \\
+W_I=\int^p_\infty\vec F_I\bullet \vec{dl}
+$$
+
+Because the desired force acts opposite to the force from the electric field, as long as $\vec E$ is known at each point:
+
+$$V_p=-\int^p_\infty\vec E\bullet\vec{dl}$$
+
+The work done only depends on initial and final positions — it is conservative, thus implying Kirchoff's voltage law.
+
+Where $\vec dl$ is the path of the test charge and $\vec dr$ is the direct path from infinity through the point to the charge, because $dr=|dl|\cos\theta$:
+
+$$\vec E\bullet\vec{dl}=Edr$$