ece106: add magnets

docs/1b/ece106.md

@@ -517,3 +517,108 @@ Much like VIR, it's usually easier to work with the form of the equation that ha
 $$U_e=\frac 1 2 \frac {Q^2}{C}=\frac 1 2 QV$$
 
 Adding dielectrics increases capacitance but decrease stored energy.
+
+## Magnetism
+
+All magnetic field lines are closed, i.e., they all return to the same magnetic object, much like a dipole. All lines must be perpendicular to the surface:
+
+$$\oint\vec B\bullet\vec{dS}=0$$
+
+Per **Biot-Savart's law**, magnets are complicated.
+
+$$\boxed{d\vec B_p=\frac{\mu_0}{4\pi}I\frac{\vec {dl}\times\hat r}{|r|^2}}$$
+
+where:
+
+- $\mu_0$ is the magnetic permeability of free space
+- $\hat r$ is the unit vector pointing from an arbitrary point of a wire to the desired point
+- $I$ is current
+- $dl$ follows the direction of current
+
+The final direction can be determined in advance with the **right-hand rule**. Therefore, magnitude can be reduced to:
+
+$$|dl\times\hat r|=|dl||\hat r|\sin\theta=|dl|\sin\theta$$
+
+### Calculations
+
+1. Define coordinate system
+2. Go to some arbitrary point $A$ on a coordinate axis such that $r=AP$
+3. Determine magnitude of the cross product
+4. Determine final magnetic field direction (should be constant)
+5. Rewrite equation in terms of one variable (usually $\theta$)
+6. Integrate
+
+### Selenoids
+
+It's easiest to place the origin at the target point.
+
+A selenoid with $N$ turns around a coil of length $L$ has density $n$, and has parallel electric fields inside.
+
+$$n=\frac N L$$
+
+The effective current of a selenoid for magnetic purposes is the sum of all currents.
+
+$$\boxed{I_{eff}=ndzI}$$
+
+where:
+
+- $dz$ is the axis in the direction of current
+- $I$ is current
+
+This can be substituted directly into Biot-Savart's law, although definite integration should be done **in the direction of the axis** (from the desired point to the farthest point of the selenoid).
+
+### Velocity and current
+
+Biot-Savart's law can be applied to moving charges:
+
+$$I\cdot \vec{dl}=\frac{dq\cdot dl}{dt}=dq\cdot \vec v$$
+
+### Ampere's law
+
+!!! definition
+    - **Drift velocity** is the average speed of electrons through a material.
+
+The **current density** $\vec J$ is the amount of charge per unit time that flows through a unit area of a cross section.
+
+$$\boxed{\vec J=nq\vec u=\rho_v\vec u}$$
+
+where:
+
+- $\vec u$ is drift velocity
+- $n$ is the charge per unit volume
+- $q$ is the total charge
+
+Ohmic resistors have current density proportional to electric field by a material's **conductivity** $\sigma$.
+
+$$\vec J=\sigma\vec E$$
+
+Resistivity is related to conductivity: $\rho=\frac 1\sigma$
+
+Integrating over a cross section returns current:
+
+$$\boxed{I=\oint\vec J\bullet\vec{dS}}$$
+
+**Ampere's law** asserts that magnetic flux due to all currents is equal to current enclosed inside a closed boundary/loop.
+
+$$
+\boxed{\begin{align*}
+\oint\vec B\bullet\vec{dl}&=\mu_0I_{enc} \\
+&=\mu_0\oint\vec J\bullet\vec{dS}
+\end{align*}}
+$$
+
+where:
+
+- $dl$ is the line along the loop/boundary in an arbitrary direction
+- $I_{enc}$ is the sum of all enclosed currents
+
+$dl$ (along the loop) and $dS$ are related in direction with each other per the **right hand rule**.
+
+For each enclosed $I$, if its direction is:
+
+- the same as $\vec dS$, it is positive in the sum term
+- opposite $\vec dS$, it is negative in the sum term
+
+1. Use $dl$ to find $dS$ or vice versa
+2. Determine $I_{enc}$
+3. Solve