ece106: i'm gonna cry

docs/1b/ece106.md

@@ -248,3 +248,35 @@ This implies $\phi_e>0$ is a net positive charge enclosed.
 
 !!! warning
     Gauss's law only applies when $\vec E$ is from all charges in the system
+
+### Charge distributed over a line/cylinder
+
+!!! warning "Limitations"
+    To apply this strategy, the following conditions must hold:
+    
+    - $Q$ must not vary with the length of the cylinder or $\phi$
+    - The charge must be distributed over either a cylindrical surface or the volume of the cylinder.
+    - In the real world, $r$ must be significantly smaller than $L$ as an approximation.
+    - The strategy is more accurate for points closer to the centre of the wire.
+
+Please see [Maxwell's integral equations#Gauss's law](https://en.wikiversity.org/wiki/MyOpenMath/Solutions/Maxwell%27s_integral_equations) for more information.
+
+**Outside** the radius $R$ of the cylinder of the Gaussian surface, the enclosed charge is, where $L$ is the length of the cylinder:
+
+$$Q_{enc}=\pi R^2\rho_0L$
+
+such that the field at any radius $r>R$ is equal to:
+
+$$\vec E(r)=\frac{\rho_0\pi R^2}{2\pi\epsilon_0r}\hat r$$
+
+**Inside** the radius $R$ of the cylinder, the enclosed charge depends on $r$. For a uniform charge density:
+
+$$Q_{enc}=\pi r^2\rho_0L$$
+
+such that the field at any radius $r< R$ is equal to:
+
+$$\vec E(r)=\frac{\rho_0}{2\epsilon_0}r\hat r$$
+
+The direction of $\vec E$ should always be equal to that of $\vec r$. Generally, where $lim$ is $r$ if $r$ is *inside* the cylinder or $R$ otherwise, $\rho_v$ is the function for charge density based on radius, and $r_1$ is hell if I know:
+
+$$\epsilon_0 E2\pi rL=\int^{lim}_0\rho_v(r_1)2\pi r_1L\ dr_1$$