ece106: add plane distribution

docs/1b/ece106.md

@@ -280,3 +280,33 @@ $$\vec E(r)=\frac{\rho_0}{2\epsilon_0}r\hat r$$
 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:
 
 $$\epsilon_0 E2\pi rL=\int^{lim}_0\rho_v(r_1)2\pi r_1L\ dr_1$$
+
+### Charge distributed over a plane
+
+!!! warning
+    To apply this strategy, the following conditions must hold:
+    
+    - $Q$ must not vary with the lengths of the plane
+    - The charge must be distributed over a plane or slab
+    - In the real world, the thickness $z$ must be significantly smaller than the lengths as an approximation
+
+Where $\rho_v$ is an **even** surface density function and $lim$ is from $0$ to $z$ if the desired field is outside of the charge, or $0$ to field height $h$ if it is inside the charge:
+
+$$\epsilon_0 E=\int_{lim}\rho_v\ dh_1$$
+
+Any two points have equal electric fields regardless of distance due to the construction of a uniform electric field.
+
+Where $\rho_v$ is not an even surface density function, $d$ is the thickness of the slab, and $E$ is the electric field **outside** the slab:
+
+$$2\epsilon_0E = \int^d_0\rho_v(A)dh_1$$
+
+Where $E$ is the electric field **inside** the slab at some height $z$:
+
+$$E=\frac{\rho_0}{4\epsilon_0}(2z^2-d^2),0\leq z\leq d$$
+
+If $E$ is negative, it must point opposite the original direction ($\hat z$).
+
+Generally:
+
+1. Determine $\vec E$ outside the slab.
+2. Set one outside surface and one inside surface as a pillbox and apply rules.