Electricity & Magnetism

I learned about electricity and magnetism in spring quarter of freshman year in Physics 43: Electricity and Magnetism

Link to Openstax textbook

Electric field energy density $u_E=\frac12\epsilon_0E^2$ :: $J/m^3$
Magnetic field energy density $u_B=\frac12\frac{B^2}{\mu_0}$
Lorentz force $\vec F=q\vec E+q\vec v\times\vec B$
Maxwell's Equations
- $∯\vec E\cdot d\vec A=\frac{Q_\text{encl.}}{\epsilon_0}$ $\iint E \cdot d A = \frac{Q _{encl.}}{ϵ _{0}}$ Gauss' Law
  - LHS is the electric flux $\Phi_E$
- $∯ \vec B\cdot d\vec A=0$ $\iint B \cdot d A = 0$ Gauss' Law for magnetism
  - In other words, there are no magnetic monopoles
- $\oint \vec E\cdot d\vec l=-\frac{d\Phi_B}{dt}$ $\oint E \cdot d l = - \frac{d Φ _{B}}{d t}$ Faraday's Law of Induction
  - A changing magnetic field produces an e-field
- $\oint \vec B\cdot d\vec l=\mu_0(I_{\text{encl.}}+I_d)$ $\oint B \cdot d l = μ_{0} (I_{encl.} + I_{d})$ Ampère–Maxwell Law where displacement current $I_d=\epsilon_0\frac{d\Phi_E}{dt}$ $I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t}$ results from a changing E-field
  - A changing e-field produces a magnetic field
Circuits
- Time constant τ tells you how long it takes for the current to reach 63.2% of its final value
Right-hand rules
- Current-carrying wire's magnetic field: point thumb in wire current's direction, fingers curl around in the direction of the B-field lines
- Solenoid: curl fingers in the direction of the current in the solenoid, thumb points in direction of B-field
- For cross-product in $\vec F=q\vec v\times \vec b$ (force on particle from B-field) and $\vec F=Id\vec l\times B$ (force on a wire with current I from a B-field)

Equation Sheet

Class Equation Sheet

Magnetism

irradiance - energy÷area received on surface ( $\frac W{m^2}$ $\frac{W}{m ^{2}}$ )
- AKA radiant flux density
- solar constant - 1 kW/m³
luminosity - rate of emission of radiation :: W
- 3E26 is solar luminosity
Electromagnetic waves
- Not exactly understood but taken for granted
  - $E(x)=cB(x)$ in a vacuum
  - energy density $u=\epsilon_0E^2$
  - $c=\frac1{\sqrt {\mu_0\epsilon_0}}$
  - Time average operator <>
  - $<S'>=\frac12\frac{E_0^2}{\mu_0c}$
- $A\cos(kx-\omega t)$ $A cos (k x - ω t)$
  - wave number $k=\frac {2\pi} \omega$
- Poynting vector $\vec S=\frac 1 {\mu_0}\vec E\times\vec B$ $S = \frac{1}{μ _{0}} E \times B$ - instantaneous rate of energy transfer per area per time (energy/area/time), power/area
  - Intensity $I = <S>$ is the time-averaged value of the Poynting vector
- Radiation pressure $p={<S> \over c}=\frac I c=u$ $p = \frac{< S >}{c} = \frac{I}{c} = u$
  - It is $2u$ if reflecting
- pressure - force per area
- intensity - power per area
Waves in general
- Period T - time taking for a wave to complete one full cycle (crest to crest or trough to trough)
- Frequency :: Hz = $s^{-1}$ - $\frac1T$ how many waves pass through a point for unit time
- Wavelength $\lambda$ - length of the wave
- $v=\frac\lambda{T}$
An electron has a magnetic dipole of 1 Bohr magneton
$B_\text{field inside}=KB_\text{vacuum}$ $B_{field inside} = K B_{vacuum}$
- A material's relative magnetic permeability $K=\mu_r =1+\chi$ $K = μ_{r} = 1 + χ$
  - $\mu_R=\frac\mu{\mu_0}$ where $\mu$ is the permeability of the material and $\mu_0$ is the permeability of free space, which is $4\pi\times10^{-7}$
  - <1 for diamagnetic
  - >1 for paramagnetic
  - $\gg1$ for ferromagnetic
- Magnetic susceptibility $\chi$
Types of magnetism
- Diamagnetism - repel, temporary
  - Material create weak repulsive force when placed in a magnetic field because the atoms' paired electrons have an induced current which changes their magnetic moments
  - Always there for all materials
  - Diamagnet: water, levitatingSoft frogs
- Paramagnetism - attract, temporary
  - Materials are weakly attracted. Too weak to pick up the material against its weight.
  - Atoms are magnetic because of unpaired electrons but in random orientation. Putting them in a magnetic field aligns them, so they are attracted toward the magnet
  - Paramagnet: aluminum
- Ferromagnetism - attract
  - Magnetic domains (particles within a domain have the same alignment) which get aligned
  - Magnetize - make the domains align
  - Hard magnets - without external magnet, stay aligned
  - Soft magnets - do not retain magnetization when removed
  - Curie temperature - the temperature at which ferromagnetism stops working because there is too much heat energy which the magnetic alignment forces can't overcome
  - Ferromagnet: iron, cobalt, nickel
Terminal voltage
- More current drawn → lower terminal voltage
Battery EMF - internal voltage inside the battery, which includes an internal resistor