diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 97fd1c8..15f5065 100644 --- a/docs/1b/ece106.md +++ b/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