Electricity & Magnetism

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

Link to OpenStax textbook

• $v(t)=v(\infty)+[v(0^+)-v(\infty)]e^{(-t/\tau)}$ works for charging and recharging RC circuit

• 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

  • $∯\vec E\cdot d\vec A=\frac{Q_\text{encl.}}{\epsilon_0}$ Gauss' Law

    • LHS is the electric flux $\Phi_E$

  • $∯ \vec B\cdot d\vec A=0$ Gauss' Law for magnetism

    • In other words, there are no magnetic monopoles

  • $\oint \vec E\cdot d\vec l=-\frac{d\Phi_B}{dt}$ Faraday's Law of Induction

    • A changing magnetic field produces an e-field

  • $\oint \vec B\cdot d\vec l=\mu_0(I_{\text{encl.}}+I_d)$ Ampère–Maxwell Law where displacement current $I_d=\epsilon_0\frac{d\Phi_E}{dt}$ 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 ($\frac W{m^2}$)

  • 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 <>

    • $<S'>=\frac12\frac{E_0^2}{\mu_0c}$

  • $A\cos(kx-\omega t)$

    • wave number $k=\frac {2\pi} \omega$

  • Poynting vector $\vec S=\frac 1 {\mu_0}\vec E\times\vec B$ - 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 $p={<S> \over c}=\frac I c=u$

    • 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

• $B_\text{field inside}=KB_\text{vacuum}$

  • A material's relative magnetic permeability $K=\mu_r =1+\chi$

    • $\mu_R=\frac\mu{\mu_0}$ 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

Dielectrics

Derivations

Deriving LR Circuit

Charging

• LR circuits

  • time constant $\tau_L=\frac L R$

  • energy stored $U_L=\frac12 LI^2$

Discharging

Deriving LRC Circuits

Deriving RC Circuits

Charging

Discharging

Deriving $\vec F=I(d\vec l\times \vec B)$

Deriving Radius of Charge in Magnetic Field

Deriving Capacitance for Series & Parallel Capacitors

Deriving Resistance for Series & Parallel Resistors