Electromagnetic Waves

Electromagnetic waves are self-propagating waves of electric and magnetic fields. They require no medium and travel at the speed of light.

Overview

graph TD
    A[EM Waves] --> B[Displacement Current]
    A --> C[Wave Properties]
    A --> D[EM Spectrum]
    C --> C1[E and B fields]
    C --> C2[Energy & Intensity]
    D --> D1[Radio to Gamma]
    D --> D2[Applications]

Displacement Current

Maxwell’s correction to Ampere’s law:

$$I_D = \varepsilon_0 \frac{d\Phi_E}{dt}$$

Modified Ampere’s Law:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0(I_c + I_D) = \mu_0 I_c + \mu_0\varepsilon_0\frac{d\Phi_E}{dt}$$

This allows magnetic fields to exist even in gaps between conductors (like in a capacitor).

Properties of EM Waves

Speed of Light

$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3 \times 10^8 \text{ m/s}$$

In a medium:

$$v = \frac{c}{n} = \frac{1}{\sqrt{\mu\varepsilon}}$$

Wave Equation

$$\vec{E} = E_0 \sin(kx - \omega t)\hat{j}$$ $$\vec{B} = B_0 \sin(kx - \omega t)\hat{k}$$

Relationships:

• $E$ and $B$ are perpendicular to each other and to propagation direction
• $\frac{E_0}{B_0} = c$
• $E$ and $B$ are in phase

Transverse Nature

EM waves are transverse waves:

• Electric field oscillates perpendicular to propagation
• Magnetic field oscillates perpendicular to both E and propagation

Energy in EM Waves

Energy Density

Electric field:

$$u_E = \frac{1}{2}\varepsilon_0 E^2$$

Magnetic field:

$$u_B = \frac{B^2}{2\mu_0}$$

Total:

$$u = u_E + u_B = \varepsilon_0 E^2 = \frac{B^2}{\mu_0}$$

Note: $u_E = u_B$ (energy equally distributed)

Intensity

$$I = \frac{1}{2}\varepsilon_0 c E_0^2 = \frac{E_0^2}{2\mu_0 c}$$

Poynting Vector

$$\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$$

Represents energy flow per unit area per unit time.

Average intensity: $I = \langle S \rangle = \frac{E_0 B_0}{2\mu_0}$

Electromagnetic Spectrum

Type	Wavelength	Frequency	Source
Radio waves	> 0.1 m	< 3 GHz	Oscillating circuits
Microwaves	0.1 m - 1 mm	3 GHz - 300 GHz	Magnetron, klystron
Infrared	1 mm - 700 nm	300 GHz - 430 THz	Hot objects
Visible	700 - 400 nm	430 - 750 THz	Excited atoms
Ultraviolet	400 - 10 nm	750 THz - 30 PHz	Sun, discharge lamps
X-rays	10 - 0.01 nm	30 PHz - 30 EHz	X-ray tubes
Gamma rays	< 0.01 nm	> 30 EHz	Nuclear reactions

Applications

Wave Type	Applications
Radio	Communication, broadcasting
Microwave	Cooking, radar, satellite
Infrared	Night vision, remote controls
Visible	Vision, photography
UV	Sterilization, vitamin D synthesis
X-rays	Medical imaging, security
Gamma	Cancer treatment, sterilization

Visible Spectrum

VIBGYOR: Violet (400 nm) → Red (700 nm)

Color	Wavelength (nm)
Violet	380-450
Blue	450-495
Green	495-570
Yellow	570-590
Orange	590-620
Red	620-750

Practice Problems

1. An EM wave has electric field amplitude 300 V/m. Find the magnetic field amplitude and intensity.

2. The electric field of an EM wave is given by $E = 100\sin(5 \times 10^{14}t - kx)$ V/m. Find wavelength and wave number.

3. A radio station broadcasts at 1000 kHz. Find the wavelength of the radio waves.

Further Reading

• Optics - Behavior of light
• Dual Nature of Matter - Particle nature of light