ece106: add faraday's law

docs/1b/ece106.md

@@ -639,3 +639,30 @@ The geometries that work include:
 2. Choose $dl$ in the direction of $B$ (counterclockwise)
 3. Determine $dS$ (out of the page) and apply Ampere's law
 
+$$\hat\phi=\hat z\times\hat r_1$$
+
+!!! warning
+    A spinning cylinder rotates faster along its outer ring, forcing an integral setup.
+   
+### Faraday's law
+
+Faraday's law states relates magnetic flux similarly to electric flux. Where $s$ is the open surface bounded by the conductor:
+
+$$\phi_m=\int_s\vec B\bullet\vec{dS}$$
+
+A flux that changes with time results in an **induced voltage** across the terminals of the conductor. Per Faraday's law of electromagnetic induction, magnetic energy is convertible to electric energy.
+
+$$V_{ind}=-\frac{d}{dt}\phi_m$$
+
+As the electric field is always perpendicular to a magnetic field, this indicates that it will curl around a straight magnetic field.
+
+Relating $dl$ and $dS$ with the right-hand rule accounts for **Lenz's law**.
+
+$$\boxed{\oint\vec E\bullet\vec{d\ell}=\frac{d}{dt}\int\vec B\bullet\vec{dS}}$$
+
+
+If there is a conducting loop in a time-varying magnetic field, a $V_{ind}$ is formed such that the current is in the direction of the induced field:
+
+$$V_{ind}=\oint\vec E\bullet\vec{d\ell}=-\frac{d}{dt}\int\vec B\bullet\vec{dS}$$
+
+Time-varying magnetic fields are formed if the field or charge is moving or if bounds change.