ece106: add brenda

docs/1b/ece106.md

@@ -656,15 +656,46 @@ $$V_{ind}=-\frac{d}{dt}\phi_m$$
 
 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**.
+Relating $dl$ and $dS$ with the right-hand rule accounts for **Lenz's law**, which creates a $\vec E$ to create a $\vec B$ to oppose the change in $\phi_m$ that created the current.
 
 $$\boxed{\oint\vec E\bullet\vec{d\ell}=\frac{d}{dt}\int\vec B\bullet\vec{dS}}$$
 
-
 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.
 
+## Inductance
 
+Kirchoff's voltage law is a simplification of Faraday's law, valid when there is no fluctuating magnetic field within the closed loop, so it's used with low frequency waves with less time variation.
+
+The **inductance** is the flux travelling through a medium over its current.
+
+$$L=\frac{\phi_m}{i}$$
+
+If there are $N$ loops in a selenoid, where $\Lambda=N\phi_m$ is the total flux/**flux linkage**, $i$ is the current in one loop, and $I$ is the current of all loops:
+
+$$L=\frac{\phi_m}{i}=\frac{\Lambda}{I_{eff}}$$
+
+The **energy density** per unit volume is $u_m$.
+
+$$u_m=\frac 1 2 \frac {B^2}{\mu_0}$$
+
+The **total work** $U_m$ done to charge current from $0$ to $I$ is related to energy density.
+
+$$U_m=\sqrt u_m=\frac 1 2 LI^2$$
+
+$$\boxed{\frac 1 2 LI^2=\frac 1 2\int_{volume}U_mdV}$$
+
+### Self-inductance
+
+A magnetic flux that passes through the current that created it will induce voltage if $I$ changes.
+
+**Mutual inductance** is wireless charging as changing current in one coil produces a changing magnetic flux in another, creating a voltage $\epsilon_{1\to 2}$.
+
+$$V_{ind}=\epsilon_{1\to 2}=N_2\frac{d\phi_{1\to 2}}{dt}=-\frac{d}{dt}\int \vec B_1\bullet\vec{dS}_2$$
+
+The mutual inductance is the rate of change of magnetic flux proportional to the rate of change of current. It is equal regardless of direction.
+
+$$\boxed{M_{1\to 2}=\frac{N_2\phi_{1\to 2}}{I_1}}$$