ece106: add point charge stuffs

docs/1b/ece106.md

@@ -322,10 +322,29 @@ $$
 
 Because the desired force acts opposite to the force from the electric field, as long as $\vec E$ is known at each point:
 
-$$V_p=-\int^p_\infty\vec E\bullet\vec{dl}$$
+$$
+V_p=-\int^p_\infty\vec E\bullet\vec{dl}
+V_p=-\int^p_\infty E\ dr
+$$
 
 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 $dr=|dl|\cos\theta$:
+Where $\vec dl$ is the path of the test charge from infinity to the point, and $\vec dr$ is the direct path from the origin through the point to the charge, because $dr=-dl$:
 
 $$\vec E\bullet\vec{dl}=Edr$$
+
+Therefore, the potential due to a point charge is equal to:
+
+$$V_p=-\int^p_\infty\frac{kQ}{r^2}dr=\frac{kQ}{r}$$
+
+**Positive** charges naturally move to **lower** potentials ($V$ decreases) while negative charges do the opposite.
+
+In order to calculate the voltage for charge distributions:
+
+- If $\vec E$ is easy to find via Gauss law:
+
+$$V_p=-\int^p_\infty\vec E\bullet\vec{dl}$$
+
+- If the charge is asymmetric:
+
+$$V_p=\int_\text{charge dist}\frac{kdQ}{r}$$