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    Electromagnetism

    Electric fields and Magnetic fields are linked to each other.

    EM Induction

    Magnetic fields exert force on moving charges; in the right conditions, this process can cause current to flow in a loop.

    Magnetic flux through a surface A (direction of A is normal to the surface):
    $$ \Phi_B=B_\parallel A \\ $$
    Changing flux in a loop of N turns induces EMF:
    $$ \begin{aligned}&{\text{In general:}} && \mathcal{E}=-N\frac{\Delta \Phi_B}{\Delta t} \\ &{\text{When area is constant:}} && \mathcal{E}=-NA\frac{\Delta B_\parallel}{\Delta t} \\ &{\text{When field is constant:}} && \mathcal{E}=-NB_\parallel\frac{ \Delta A}{\Delta t} \\ &{\text{When loop rotates:}} && |\mathcal{E}|=\omega N A B \sin{(\omega t)} \\ \end{aligned} $$

    Angular frequency

    Rotation frequency can be represented in Hertz (cycles per second, f) or in radians per second (ω). The conversion between the two is ω=2πf

    Transformers

    In a transformer, a primary coil (P) with NP turns is driven by a power source, causing emf to be induced in a secondary coil (S) with NS turns.

    Voltage, current, and turn relationships in a transformer
    $$ \frac{V_P}{V_S}=\frac{I_S}{I_P}=\frac{N_P}{N_S} \\ $$

    RMS values

    Alternating Current signals are described using Root Mean Squared (RMS) values. These values represent the equivalent constant Direct Current (DC) voltage for the same amount of power used.

    RMS Voltage and Current
    $$ \begin{aligned}&{\text{Voltage:}} && V_{rms}={\textstyle\frac{1}{\sqrt{2}}}V_{max} \\ &{\text{Current:}} && I_{rms}={\textstyle\frac{1}{\sqrt{2}}}I_{max} \\ \end{aligned} $$

    Electromagnetic Waves

    Electromagnetic waves involves the simultaneous "waving" of electric fields and magnetic fields as they propagate through space.

    Wave properties #properties
    $$ \begin{aligned}&{\text{Propagation velocity}}&& v \\&{\text{Frequency}}&& f \\&{\text{Wavelength}}&& \lambda \\&{\text{Power (Usually from Source)}}&& P \\&{\text{Intensity}}&& I \text{ or }S \\&{\text{Energy density}}&& u \end{aligned} $$
    Relationship between wavelength, frequency, and wave speed:
    $$ v=\lambda f $$

    EM Wave Properties

    Speed of EM waves in vacuum
    $$ c=\frac{1}{\sqrt{\varepsilon_0 \mu_0}} =3\times 10^8 \mbox{ m/s} $$
    Electric and Magnetic Field magnitude relationship
    $$ |E|=c|B| $$

    EM Wave Power and Intensity

    Electric and Magnetic fields in an EM wave are not constant. For this reason, we describe energy transmission with EM waves using time-averaged values.

    Symbols to indicate a value is time-averaged
    $$ \begin{aligned}&{\text{Bracket around variable}}&& \langle x \rangle \\&{\text{Line over variable}}&& \overline{x} \end{aligned} $$
    Wave Intensity (S or I) measures average wave power per unit area at a given location:
    $$ S=I=\frac{\langle P \rangle}{A} $$

    EM Wave Energy Density

    Time-averaged total energy density is related to intensity:
    $$ S=I=\langle u_{total} \rangle c $$
    Electric field and Magnetic field carry energy distributed with energy density:
    $$ \begin{aligned}&{\text{Electric energy density:}}&& u_E={\textstyle \frac{1}{2}}\varepsilon_0 E^2 \\&{\text{Magnetic energy density:}}&& u_B={\textstyle \frac{1}{2 \mu_0}} B^2 \end{aligned} $$
    Replace field magnitudes with Root Mean Square (rms) values to get time-averaged energy densities:
    $$ \begin{aligned}&{\text{Electric avg energy density:}}&& \langle u_E \rangle={\textstyle \frac{1}{2}}\varepsilon_0 E_{rms}^2 \\&{\text{Magnetic avg energy density:}}&& \langle u_B \rangle={\textstyle \frac{1}{2 \mu_0}} B_{rms}^2 \end{aligned} $$

    rms values

    For any sinusoidal signal with maximum value Amax, the root mean square value is Arms=Amax/√2

    Total energy density is the sum of Electric and Magnetic field energy densities
    $$ \begin{aligned}\langle u_{total} \rangle &={\langle u_{E} \rangle + \langle u_{B} \rangle} \\ &={{\textstyle \frac{1}{2}}\varepsilon_0 E^2+{\textstyle \frac{1}{2 \mu_0}} B_{rms}^2} \quad \text{Use} \; E=cB \text{ to combine terms}\\ &={\varepsilon_0 E_{rms}^2} \\ &={\textstyle \frac{1}{\mu_0}} B_{rms}^2\end{aligned} $$

    Electromagnetism constants

    Permittivity of free space:
    ε0=8.85×10-12 C2/N‧m2

    Vacuum magnetic permeability:
    μ0=4π×10-7 T‧m/A2

    These two properties of vacuum allow EM waves to travel at the speed of light:
    c=1/√(ε0μ0)