Electromagnetism

As discussed in EMF, magnetic and electric fields are interchangeable with the right context and one cannot exist without the other.

Maxwell Equations in a Vacuum

LAW	INTEGRAL FORM	DIFFERENTIAL FORM
GAUSS FOR ELECTRIC FIELD	${\oint }_S\vec{E}(\vec{r})\cdot {\vec{u}}_{N}dS=\frac{Q}{{ϵ}_0}$	$\nabla \cdot \vec{E}=\frac{\rho }{{ϵ}_0}$
GAUSS FOR MAGNETIC FIELD	${\oint }_S\vec{B}(\vec{r})\cdot {\vec{u}}_{N}dS=0$	$\nabla \cdot \vec{B}=0$
FARADAY-HENRY-LENZ	${\oint }_{\Gamma }\vec{E}(\vec{r})\cdot d\vec{s}=-\frac{d}{dt}\left[{\int }_S\vec{B}(\vec{r})\cdot {\vec{u}}_{N}dS\right]$	$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$
AMPERE-MAXWELL	${\oint }_{\Gamma }\vec{B}(\vec{r})\cdot d\vec{s}={\mu }_{0}I+{\mu }_0{ϵ}_0\frac{d\Phi (\vec{E})}{dt}$	$\nabla \times \vec{B}={\mu }_0\vec{J}+{\mu }_0{ϵ}_0\frac{\partial \vec{E}}{\partial t}$

EM Waves

Accelerating charges create self-propagating $E$ and $B$ fields.

EM Properties (In Vacuum)

• Speed: $v_{em}=c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\approx 3\times 10^8{\text{ ms}}^{-1}$
• Geometry: $\vec{E}\perp \vec{B}\perp \vec{k}$ (Transverse wave).
• Magnitude: $|\vec{E}|=c|\vec{B}|$ (Electric field is much stronger).
• Vector Relation: $\vec{k}\times \vec{E}=\omega \vec{B}$

The Poynting Vector ( $\vec{S}$)

Represents the energy flux (energy transfer per unit area per unit time).

$\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B}$

Simplified magnitude: $|\vec{S}|=c{\epsilon }_0|\vec{E}{|}^2$

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Light & Conservation of Energy

Light carries intensity ($I$). When it hits a surface, it splits into three fates: Transmitted ($T$), Reflected ($R$), or Absorbed ($A$).

Conservation Law

Since energy is conserved: $I_0=I_T+I_R+I_A$ Dividing by initial intensity $I_0$: $1=T+R+A$

Refractive Index ( $n$)

In a material, light slows down.

• Speed: $v=\frac{c}{n}$
• Wavelength: $\lambda =\frac{{\lambda }_0}{n}$ (Frequency $f$ never changes!)
• Wavenumber: $k=n{k}_0$

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Absorption (Beer-Lambert Law)

As light travels through a thick medium, it gets eaten up.

Beer-Lambert Law

Intensity decays exponentially with distance $x$: $I(x)=I_0{e}^{-\alpha x}$

• $\alpha$: Absorption coefficient (depends on material).
• $\kappa$: Extinction coefficient (linked to complex refractive index $N=n+i\kappa$). $\alpha =\frac{4\pi \kappa }{{\lambda }_0}$

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Reflection & Refraction (Geometric Optics)

How light bends and bounces.

Snell's Law (Refraction)

When moving between media with different indices ($n_i$ to $n_t$): $n_i\sin {\theta }_i=n_t\sin {\theta }_t$

• $\theta$ is always measured from the Normal (vertical axis).

Fermat's Principle

Light always takes the path of least time (not necessarily shortest distance).

• Mirages: Caused by hot air changing $n$, bending light so the “fastest” path curves through the sky.
• Reflection: ${\theta }_{incident}={\theta }_{reflected}$

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Material Behaviors

Why mirrors shine and glass is clear.

Metals vs. Dielectrics

Metals (High $\kappa$):

• High absorption coefficient means light is absorbed instantly at the surface.
• This energy vibrates electrons, which immediately re-emit the light.
• Result: High Reflection ($R\approx 1$).

Glass (Dielectric):

• $R\approx 0.04$ (Only ~4% reflected per surface).
• Most light is transmitted ($T$).

Total Internal Reflection

If you try to go from High $n$ to Low $n$ (e.g., Water to Air) at a steep angle, light gets trapped.

• Happens when ${\theta }_i>{\theta }_{critical}$.

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Beatings

When two waves of slightly different frequencies overlap, they create a pulsing effect.

The Envelope Effect

The total wave looks like a high-frequency “carrier” wave inside a low-frequency “envelope.”

• Carrier Frequency: Average of the two ($\frac{{\omega }_1+{\omega }_2}{2}$).
• Modulator (Beat) Frequency: Difference of the two ($\frac{{\omega }_1-{\omega }_2}{2}$).

$A_{tot}(x,t)=\left\{2A_0\cos \left[\frac{\Delta k}{2}x-\frac{\Delta \omega }{2}t\right]\right\}\sin [kx-\omega t]$

Application: AM Radio (Amplitude Modulation) uses this to encode sound onto a radio wave.

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Interference (Two Sources)

When two coherent waves (same frequency, constant phase difference) meet.

Constructive vs. Destructive

• Phase Difference ($\Delta \varphi$): Depends on the path difference ($\Delta s$) traveled by the waves. $\Delta \varphi =\frac{2\pi }{\lambda }\Delta s$
• Constructive ($0,2\pi ,\dots$): Waves add up. Peak intensity $\propto 4A_0^2$.
• Destructive ($\pi ,3\pi ,\dots$): Waves cancel out. Intensity is zero.

Young's Double Slit

For two slits separated by distance $d$:

• Path Difference: $\Delta s\approx d\sin \theta$
• Maxima (Bright Spots): $\sin \theta =m\frac{\lambda }{d}$ (where $m$ is an integer).
• Minima (Dark Spots): $\sin \theta =(m+\frac{1}{2})\frac{\lambda }{d}$

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N-Slit Interference

What happens when you add more slits ($N>2$).

Sharper Peaks

As $N$ increases:

• Brightness explodes: Max Intensity $\propto N^2$.
• Sharpness increases: Peak width $\propto 1/N$.
• Secondary Peaks: Small ripples appear between the main bright spots.

This is the principle behind Phased Array Antennas - you can steer a signal beam just by changing the timing (phase) of the emission, without physically moving the antenna.

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Diffraction

The bending of light when it passes through a single small opening (width $a$).

The Spreading Limit

If the hole is smaller than the wavelength ($a<\lambda$), light spreads everywhere spherically. If the hole is large, you get a distinct pattern.

Fraunhofer Diffraction (Long distance approximation): $I(\theta )=I_0{\text{sinc}}^2\alpha \text{where}\alpha =\frac{\pi a\sin \theta }{\lambda }$

Diffraction Minima (Dark Spots)

The condition for DARK spots in a single slit is: $\sin {\theta }_{min}=n\frac{\lambda }{a}(n\ne 0)$

• Crucial Note: $n=0$ is the Central Maximum (the big bright spot in the middle).
• The central spot is twice as wide as the other fringes.