diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 3bab104..d2d0463 100644 --- a/docs/1b/ece106.md +++ b/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.