ece106: add electrostatics and moment

docs/1b/ece106.md

@@ -49,10 +49,9 @@ These rules also apply for a system in three dimensions:
 
 Although differential elements can be blindly used inside and outside an object (e.g., area), the rules break down as the **boundary** of an object is approached (e.g., perimeter). Applying these rules to determine an object's perimeter will result in the incorrect deduction that $\pi=4$.
 
-Therefore, further approximations can be made by making a length $\dl=\sqrt{(dx)^2+(dy)^2}$ to represent the perimeter.
+Therefore, further approximations can be made using the Pythagorean theorem to represent the perimeter.
 
-!!! example
+$$dl=\sqrt{(dx^2) + (dy)^2}$$
-    This reduces to $dl=\sqrt{\left(\frac{dy}{dx}\right)^2+1}$.
 
 ### Polar coordinates
 
@@ -79,6 +78,15 @@ $$dS=(dr)(rd\phi)$$
     \end{align*}
     $$
 
+If $r$ does not depend on $d\phi$, part of the integral can be pre-evaluated:
+
+$$
+\begin{align*}
+dS&=\int^{2\pi}_{\phi=0} r\ dr\ d\phi \\
+dS^\text{ring}&=2\pi r\ dr
+\end{align*}
+$$
+
 So long as the variables are independent of each other, their order does not matter. Otherwise, the dependent variable must be calculated first.
 
 
@@ -90,7 +98,72 @@ So long as the variables are independent of each other, their order does not mat
     \int^b_a\sin^2\phi=\frac{b-a}{2}
     $$
 
-## Cartesian coordinates
+The side length of a curve is as follows:
 
-The axes in a Cartesian coordinate plane must be orthogonal so that increasing a value in one axis does not affect any other. The axes must also point in directions that follow the **right hand rule**.
+$$dl=\sqrt{(dr^2+(rd\phi)^2}$$
 
+!!! example
+    The side length of the curve $r=e^\phi$ (Archimedes' spiral) from $0$ to $2\pi$:
+    
+    \begin{align*}
+    dl &=d\phi\sqrt{\left(\frac{dr}{d\phi}\right)^2 + r^2} \\
+    \tag{$\frac{dr}{d\phi}=e^\phi$}&=d\phi\sqrt{e^{2\phi}+r^2} \\
+    &=????????
+    \end{align*}
+
+Polar **volume** is the same as Cartesian volume:
+
+$$dV=A\ dr$$
+
+!!! example
+    For a cylinder of radius $R$ and height $h$:
+    
+    $$
+    \begin{align*}
+    dV&=\pi R^2\ dr \\
+    V&=\int^h_0 \pi R^2\ dr \\
+    &=\pi R^2 h
+    \end{align*}
+    $$
+
+### Moment of inertia
+
+The **mass distribution** of an object varies depending on its surface density $\rho_s$. In objects with uniformly distributed mass, the surface density is equal to the total mass over the total area.
+
+$$dm=\rho_s\ dS$$
+
+The formula for the **moment of inertia** of an object is as follows, where $r_\perp$ is the distance from the axis of rotation:
+
+$$dI=(r_\perp)^2dm$$
+
+!!! example
+    In a uniformly distributed disk rotating about the origin like a CD with mass $M$ and radius $R$:
+    
+    $$
+    \begin{align*}
+    \rho_s &= \frac{M}{\pi R^2} \\
+    dm &= \rho_s\ r\ dr\ d\phi \\
+    dI &=r^2\ dm \\
+    &= r^2\rho_s r\ dr\ d\phi \\
+    &= \rho_s r^3dr\ d\phi \\
+    I &=\rho_s\int^{2\pi}_{\phi=0}\int^R_{r=0} r^3dr\ d\phi \\
+    &= \rho_s\int^{2\pi}_{\phi=0}\frac{1}{4}R^4d\phi \\
+    &= \rho_s\frac{1}{2}\pi R^4 \\
+    &= \frac 1 2 MR^2
+    \end{align*}
+    $$
+
+## Electrostatics
+
+!!! definition
+    - The **polarity** of a particle is whether it is positive or negative.
+
+The law of **conservation of charge** states that electrons and charges cannot be created nor destroyed, such that the **net charge in a closed system stays the same**.
+
+The law of **charge quantisation** states that charge is discrete — electrons have the lowest possible quantity.
+
+Please see [SL Physics 1#Charge](/sph3u7/#charge) for more information.
+
+**Coulomb's law** states that for point charges $Q_1, Q_2$ with distance from the first to the second $\vec R_{12}$:
+
+$$\vec F_{12}=k\frac{Q_1Q_2}{||R_{12}||^2}\hat{R_{12}}$$

docs/1b/math119.md

@@ -56,7 +56,7 @@ In practice, this means that if any two paths result in different limits, the li
     
     Along $y=0$:
     
-    $$\lim_{(x,0)\to(0, 0) ... = 1$$
+    $$\lim_{(x,0)\to(0, 0)} ... = 1$$
     
     Along $x=0$: