Electromagnetic Waves

ELECTROMAGNETIC WAVES

Introduction and Maxwell's Contribution

**Maxwell's Equations** unify electricity, magnetism, and light into a consistent mathematical framework. James Clerk Maxwell (1831–1879) demonstrated that:

A time-varying electric field produces a magnetic field (symmetric to Faraday's law)

The speed of electromagnetic waves equals the speed of light (3 × 10⁸ m/s)

**Light is an electromagnetic wave**

This revolutionary conclusion unified three domains of physics. Heinrich Hertz experimentally verified electromagnetic waves in 1887 (radio region, λ = 25 mm to 5 mm), confirming Maxwell's predictions and enabling modern communication technology.

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DISPLACEMENT CURRENT

The Problem with Ampere's Circuital Law

**Ampere's Circuital Law** (Chapter 4):

∮**B**·d**l** = μ₀i(t)

When applying this law to a parallel plate capacitor being charged, a **contradiction** arises:

**Outside the capacitor** (Fig. 8.1a): A current i flows through the wire, so ∮**B**·d**l** = μ₀i

**Between capacitor plates** (Fig. 8.1b and c): No conduction current passes through surfaces with the same boundary loop, yet the magnetic field **B** at point P should be the same by symmetry

This inconsistency reveals that Ampere's law is **incomplete**.

Maxwell's Solution: Displacement Current

**Definition of Displacement Current:**

The **displacement current** is a new term that arises from a time-varying electric field:

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

where:

ε₀ = permittivity of free space (8.85 × 10⁻¹² F/m)

Φ_E = electric flux through the surface

**Derivation of the Missing Term:**

Inside the parallel plate capacitor with charge Q:

Electric field: E = Q/(A·ε₀)

Electric flux through surface S: Φ_E = E·A = Q/ε₀

Since charge changes with time (Q = ∫i dt):

$$\frac{d\Phi_E}{dt} = \frac{1}{\varepsilon_0}\frac{dQ}{dt} = \frac{i}{\varepsilon_0}$$

Therefore:

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

This **displacement current** equals the conduction current, resolving the contradiction.

Physical Interpretation

**Key Distinctions:**

**Conduction current (i_c)**: Flow of actual charges through a conductor (outside capacitor plates, i_c = i)

**Displacement current (i_d)**: Due to changing electric field (between capacitor plates, i_c = 0, i_d = i)

**Total current:**

$$i_{total} = i_c + i_d = i_c + \varepsilon_0 \frac{d\Phi_E}{dt}$$

The displacement current has **identical effects** to conduction current—it generates a magnetic field.

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AMPERE-MAXWELL LAW

Generalized Ampere's Law

**Statement:** The magnetic field around a closed loop equals μ₀ times the total current (conduction + displacement) passing through any surface bounded by the loop.

**Mathematical Form:**

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \left( i_c + \varepsilon_0 \frac{d\Phi_E}{dt} \right)$$

Or equivalently:

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$$

Significance and Symmetry

**Restored Symmetry in Maxwell's Equations:**

**Faraday's Law**: Time-varying magnetic field → Electric field

$$\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$$

**Ampere-Maxwell Law**: Time-varying electric field → Magnetic field

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$$ (in region with no conduction current)

This **mutual regeneration** of time-varying E and B fields is the origin of **electromagnetic waves**.

Example Application

In a parallel plate capacitor with alternating voltage:

**Between plates**: E varies with time → displacement current i_d flows → magnetic field B is generated (perpendicular to E and circular around the axis)

**Outside plates**: Conduction current i_c flows → same magnetic field B

At the boundary: Both effects contribute, maintaining consistency

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MAXWELL'S EQUATIONS IN VACUUM

**Maxwell's four equations form the complete description of electromagnetism:**

1. **Gauss's Law (Electricity):**

$$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\varepsilon_0}$$

(Electric charges create electric fields)

2. **Gauss's Law (Magnetism):**

$$\oint \mathbf{B} \cdot d\mathbf{A} = 0$$

(No magnetic monopoles exist)

3. **Faraday's Law:**

$$\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$$

(Changing magnetic flux creates electric field)

4. **Ampere-Maxwell Law:**

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 i_c + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$$

(Current and changing electric flux create magnetic field)

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ELECTROMAGNETIC WAVES: SOURCES AND NATURE

Sources of Electromagnetic Waves

**Key Principle:** **Only accelerated charges radiate electromagnetic waves.**

**Non-sources:**

Stationary charges: Produce only static electric field (no variation)

Uniformly moving charges (steady current): Produce static magnetic field (no variation)

**Why accelerated charges?**

An oscillating (accelerating) charge creates:

1. Oscillating electric field in space

2. This E-field variation → oscillating magnetic field (via Ampere-Maxwell law)

3. This B-field variation → oscillating electric field (via Faraday's law)

4. Mutual regeneration continues as wave propagates

The wave frequency equals the **oscillation frequency** of the source charge.

**Historical Verification:**

**Hertz (1887)**: Demonstrated EM waves using oscillating LC circuits (frequency ~10⁹ Hz, λ ≈ 30 cm)

**Jagdish Chandra Bose (1895)**: Produced shorter wavelengths (5–25 mm) at Calcutta

**Marconi (1895)**: First long-distance transmission (kilometers), founding wireless communication

**Why not visible light?** Yellow light oscillates at 6 × 10¹⁴ Hz—impossible to generate with circuits (max ~10¹¹ Hz). EM waves were demonstrated first in radio region, confirming that light is also EM waves.

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PROPERTIES OF ELECTROMAGNETIC WAVES

Nature and Orientation

**Fundamental Properties:**

1. **Perpendicularity:** E, B, and direction of propagation (**k**) are **mutually perpendicular** (transverse wave)

2. **In-phase oscillation:** E and B oscillate in phase (same time, same phase)

3. **Sinusoidal variation:** For a wave propagating along +z direction:

$$E_x = E_0 \sin(kz - \omega t)$$

$$B_y = B_0 \sin(kz - \omega t)$$

where:

**E₀** = electric field amplitude (V/m)

**B₀** = magnetic field amplitude (T)

**k** = wave number = 2π/λ (rad/m)

**ω** = angular frequency = 2πν (rad/s)

**λ** = wavelength (m)

**ν** = frequency (Hz)

Wave Relations

**Dispersion Relation:**

$$\omega = ck$$

where **c** = speed of EM wave in vacuum

**Relation to frequency and wavelength:**

$$

u \lambda = c$$

or

$$c =

u \lambda = \frac{\omega}{k}$$

**Field Magnitude Relation:**

From Maxwell's equations:

$$B_0 = \frac{E_0}{c}$$

This shows **B₀ << E₀** (since c = 3 × 10⁸ m/s is large). For example, if E₀ = 6 V/m, then B₀ = 2 × 10⁻⁸ T.

Speed in Different Media

**In vacuum:**

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

**In a material medium** (with permittivity ε and permeability μ):

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

The speed is **slower** in a medium and depends on **refractive index** n = c/v (discussed in Ray Optics chapter).

Self-Sustaining Nature

EM waves are **self-sustaining oscillations** of coupled E and B fields

**No material medium required** for propagation (unlike sound or water waves)

Vacuum is a perfect medium for EM wave propagation

Energy is transported by the wave from source to distant locations

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WORKED NUMERICAL EXAMPLES

**Example 8.1: Finding Magnetic Field from Electric Field**

A plane EM wave of frequency ν = 25 MHz travels in free space along the x-direction. At a point, **E** = 6.3**ĵ** V/m. Find **B**.

**Solution:**

*Step 1: Find magnitude using B₀ = E₀/c*

$$B_0 = \frac{E_0}{c} = \frac{6.3}{3 \times 10^8} = 2.1 \times 10^{-8} \text{ T}$$

*Step 2: Find direction using **E** × **B** || direction of propagation*

Wave propagates along **x**-direction (given)

**E** is along **y**-direction (given as 6.3**ĵ**)

Need **E** × **B** = direction of propagation

For (**ĵ**) × (**B**) = (**î**):

**B** must be along **k̂** (z-direction)

Check: (**ĵ**) × (**k̂**) = (**î**) ✓

*Step 3: Final answer*

$$\boxed{\mathbf{B} = 2.1 \times 10^{-8} \hat{\mathbf{k}} \text{ T}}$$

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**Example 8.2: Finding Wavelength and Frequency from Wave Equation**

The magnetic field in a plane EM wave is:

$$B_y = (2 \times 10^{-7}) \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \text{ T}$$

(a) Find wavelength and frequency

(b) Write the electric field equation

**Solution:**

*Step 1: Extract wave parameters from equation By = B₀ sin(kx + ωt)*

Comparing with standard form B_y = B₀ sin(kz ± ωt):

**k** = 0.5 × 10³ rad/m = 500 rad/m

**ω** = 1.5 × 10¹¹ rad/s

*Step 2: Calculate wavelength*

$$\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.5 \times 10^3} = \frac{6.28}{500} = 1.26 \times 10^{-2} \text{ m} = 1.26 \text{ cm}$$

*Step 3: Calculate frequency*

$$

u = \frac{\omega}{2\pi} = \frac{1.5 \times 10^{11}}{2\pi} = \frac{1.5 \times 10^{11}}{6.28} = 2.39 \times 10^{10} \text{ Hz} = 23.9 \text{ GHz}$$

*Verification:* νλ = (2.39 × 10¹⁰)(1.26 × 10⁻²) = 3 × 10⁸ m/s = c ✓

*Step 4: Find electric field amplitude*

$$E_0 = cB_0 = (3 \times 10^8)(2 \times 10^{-7}) = 60 \text{ V/m}$$

*Step 5: Determine electric field direction and write equation*

B is along y-direction, propagates along x-direction (from the equation form)

**E** must be perpendicular to both: along z-direction

Phase: Since B_y = B₀ sin(kx + ωt), the wave travels in **–x direction**

**E** and **B** oscillate in phase

$$\boxed{E_z = 60 \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \text{ V/m}}$$

or equivalently:

$$\boxed{E_z = 60 \sin(500x + 1.5 \times 10^{11} t) \text{ V/m}}$$

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KEY EXAM POINTS TO REMEMBER

**Displacement current:** id = ε₀(dΦ_E/dt); has same effects as conduction current

**Ampere-Maxwell law** resolves the contradiction in Ampere's law by including displacement current

**EM waves** are produced only by **accelerated charges**, not stationary or uniformly moving charges

**E, B, and k** are mutually perpendicular in an EM wave

**Relation:** B₀ = E₀/c (magnetic amplitude much smaller than electric)

**Speed in vacuum:** c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s (fundamental constant)

**Wave equation:** νλ = c or ω = ck

**In media:** v = c/n = 1/√(με), where n is refractive index

**Maxwell's equations** (all four) are the foundation of classical electromagnetism

**Symmetry:** Faraday's law (B → E) and Ampere-Maxwell law (E → B) show mutual induction

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ELECTROMAGNETIC SPECTRUM (PREVIEW)

EM waves span an enormous range of wavelengths/frequencies:

**Radio waves:** λ ~ 10³–10⁶ m (ν ~ 10⁴–10¹¹ Hz)

**Microwaves:** λ ~ 10⁻³–10⁻¹ m (ν ~ 10⁹–10¹¹ Hz)

**Infrared:** λ ~ 10⁻⁷–10⁻⁴ m

**Visible light:** λ ~ 4–8 × 10⁻⁷ m (ν ~ 4–8 × 10¹⁴ Hz)

**Ultraviolet:** λ ~ 10⁻⁸–10⁻⁷ m

**X-rays:** λ ~ 10⁻¹⁰–10⁻⁸ m

**Gamma rays:** λ ~ 10⁻¹²–10⁻¹⁰ m

All travel at c in vacuum and obey the same Maxwell equations (detailed coverage in EM Spectrum section of full chapter).