Electromagnetic waves are self-propagating disturbances in electromagnetic fields. Unlike mechanical waves that require a medium, EM waves can travel through empty space because they consist of oscillating electric and magnetic fields that regenerate each other.
Imagine a changing electric field creating a changing magnetic field, which in turn creates a changing electric field, and so on. This creates a “chain reaction” that propagates through space at the speed of light.
Ampère’s original law ∮B⃗·dl⃗ = μ₀I worked for steady currents, but failed for time-varying situations like charging capacitors. Between capacitor plates, there’s no conduction current, yet magnetic fields exist!
Maxwell realized that changing electric field should produce magnetic field, just like current does.
Step 1: Start with Gauss’s law inside capacitor:
∮D⃗·dA⃗ = Q_free, where D⃗ = ε₀E⃗ + P⃗
Step 2: For linear dielectric: D⃗ = εE⃗ = ε₀εᵣE⃗
In vacuum: D⃗ = ε₀E⃗
Step 3: Take time derivative:
∂/∂t(∮D⃗·dA⃗) = dQ/dt = I
Step 4: This gives displacement current:
\[I_d = \frac{\partial}{\partial t}\int \vec{D} \cdot d\vec{A} = \epsilon_0 \frac{\partial}{\partial t}\int \vec{E} \cdot d\vec{A}\]
\[I_d = \epsilon_0 \frac{\partial \Phi_E}{\partial t}\]
It’s NOT a flow of charges! It’s the rate of change of electric field that acts like a current in producing magnetic effects. The word “displacement” refers to electric displacement field D⃗.
During charging: conduction current I flows in wires, displacement current I_d = I flows between plates. Total current is continuous!
If E = E₀sin(ωt) between plates:
I_d = ε₀A(dE/dt) = ε₀AωE₀cos(ωt)
This displacement current creates magnetic field around capacitor!
Modified Ampère’s Law:
\[\oint \vec{B} \cdot d\vec{l} = \mu_0\left(I_{enc} + I_d\right) = \mu_0\left(I_{enc} + \epsilon_0\frac{d\Phi_E}{dt}\right)\]
Faraday’s Law: ∇×E⃗ = -∂B⃗/∂t
Modified Ampère’s Law: ∇×B⃗ = μ₀ε₀∂E⃗/∂t
Step 1: Take curl of Faraday’s law:
∇×(∇×E⃗) = -∂/∂t(∇×B⃗)
Step 2: Use vector identity: ∇×(∇×E⃗) = ∇(∇·E⃗) – ∇²E⃗
Since ∇·E⃗ = 0 in vacuum: ∇×(∇×E⃗) = -∇²E⃗
Step 3: Substitute Ampère’s law:
-∇²E⃗ = -∂/∂t(μ₀ε₀∂E⃗/∂t) = -μ₀ε₀∂²E⃗/∂t²
Step 4: Final wave equation:
\[\nabla^2 \vec{E} = \mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\]
\[\nabla^2 \vec{B} = \mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2}\]
This is the standard wave equation ∇²f = (1/v²)∂²f/∂t² with speed:
v = 1/√(μ₀ε₀) = c = 2.998×10⁸ m/s
μ₀ (Permeability of free space):
ε₀ (Permittivity of free space):
Higher μ₀ and ε₀ mean the medium “resists” field changes more, making EM waves propagate slower. The speed is inversely related to how much the medium “opposes” electromagnetic field formation.
\[c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = \frac{1}{\sqrt{4\pi \times 10^{-7} \times 8.854 \times 10^{-12}}}\]
\[c = \frac{1}{\sqrt{1.112 \times 10^{-17}}} = 2.998 \times 10^8 \text{ m/s}\]
Assume solutions of form:
E⃗ = E₀f(z-ct)x̂, B⃗ = B₀f(z-ct)ŷ
Apply Maxwell’s equations:
From ∇×E⃗ = -∂B⃗/∂t:
∂Ez/∂y – ∂Ey/∂z = -∂Bx/∂t
Since Ex = E₀f(z-ct), Ey = Ez = 0:
-∂Ex/∂z = -∂By/∂t
E₀f'(z-ct) = B₀cf'(z-ct)
Therefore: E₀ = B₀c
From ∇×B⃗ = μ₀ε₀∂E⃗/∂t:
∂Bx/∂y – ∂By/∂z = μ₀ε₀∂Ex/∂t
B₀f'(z-ct) = μ₀ε₀E₀(-c)f'(z-ct)
This gives: B₀ = μ₀ε₀cE₀ = E₀/c
Key Relations for Plane Waves:
\[\frac{E_0}{B_0} = c\]
\[\vec{E} \perp \vec{B} \perp \hat{k}\]
\[E_x = E_0 \cos(kz – \omega t + \phi)\]
\[B_y = B_0 \cos(kz – \omega t + \phi) = \frac{E_0}{c} \cos(kz – \omega t + \phi)\]
Electric Field Energy Density:
u_E = ½ε₀E² comes from work done to create electric field
Think of energy stored in “electric field tension”
Magnetic Field Energy Density:
u_B = B²/(2μ₀) comes from work done to create magnetic field
Think of energy stored in “magnetic field strain”
Electric energy density:
u_E = ½ε₀E² = ½ε₀E₀²cos²(kz-ωt)
Magnetic energy density:
u_B = B²/(2μ₀) = (E₀/c)²/(2μ₀) cos²(kz-ωt)
= E₀²/(2μ₀c²) cos²(kz-ωt)
= E₀²ε₀/(2) cos²(kz-ωt) [since c² = 1/(μ₀ε₀)]
= ½ε₀E₀²cos²(kz-ωt)
Therefore: u_E = u_B at every instant!
\[u_{total} = u_E + u_B = 2u_E = \epsilon_0 E^2\]
\[\langle u \rangle = \frac{\epsilon_0 E_0^2}{2}\] (time average)
The Poynting vector S⃗ represents the rate of energy flow per unit area (power density) carried by electromagnetic fields. It points in the direction of energy propagation.
Start with energy density: u = ½(ε₀E² + B²/μ₀)
Energy conservation equation:
∂u/∂t + ∇·S⃗ = -J⃗·E⃗ (energy dissipation)
Using Maxwell’s equations and vector identities:
∇·S⃗ = ∇·(E⃗×B⃗/μ₀) = B⃗·(∇×E⃗)/μ₀ – E⃗·(∇×B⃗)/μ₀
Substituting Maxwell’s equations:
= -B⃗·∂B⃗/∂t/μ₀ – E⃗·(μ₀J⃗ + μ₀ε₀∂E⃗/∂t)/μ₀
= -∂u/∂t – J⃗·E⃗
This confirms:
\[\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})\]
Magnitude: \[S = \frac{EB}{\mu_0} = \frac{E^2}{\mu_0 c}\]
The Poynting vector oscillates rapidly (at frequency ~10¹⁴ Hz for visible light). Detectors measure the time-averaged value over many cycles, called intensity.
For plane wave: S = EB/μ₀ = E₀B₀cos²(kz-ωt)/μ₀
Time average of cos²(kz-ωt):
⟨cos²(kz-ωt)⟩ = ½ (average over complete cycles)
Therefore:
I = ⟨S⟩ = E₀B₀/(2μ₀) = E₀²/(2μ₀c) [since B₀ = E₀/c]
Alternative forms:
I = cε₀E₀²/2 = cB₀²/(2μ₀)
\[I = \frac{E_0^2}{2\mu_0 c} = \frac{c\epsilon_0 E_0^2}{2} = \frac{cB_0^2}{2\mu_0}\]
Units: W/m² (watts per square meter)
Intensity tells us how much electromagnetic energy passes through a unit area per unit time. Higher intensity means brighter light or stronger EM waves.
Even though photons are massless, they carry energy E = pc (relativistic relation for massless particles). Since EM waves are made of photons, they must carry momentum.
From special relativity: E = pc for massless particles
For EM waves: E = hf, so p = hf/c = E/c
Energy density: u = ε₀E²
Momentum density: g = u/c = ε₀E²/c
Momentum flux (momentum transfer per unit time per unit area):
= momentum density × speed = (ε₀E²/c) × c = ε₀E²
For time-averaged intensity I:
Average momentum flux = I/c
Radiation Pressure:
Complete absorption: \[P = \frac{I}{c}\]
Complete reflection: \[P = \frac{2I}{c}\]
Solar intensity at Earth ≈ 1360 W/m²
Radiation pressure ≈ 1360/(3×10⁸) ≈ 4.5×10⁻⁶ Pa
This tiny pressure pushes comet tails away from Sun!
1. Gauss’s Law for Electricity:
∮E⃗·dA⃗ = Q_enc/ε₀
Meaning: Electric charges create electric fields that diverge from positive charges and converge to negative charges.
2. Gauss’s Law for Magnetism:
∮B⃗·dA⃗ = 0
Meaning: No magnetic monopoles exist; magnetic field lines always form closed loops.
3. Faraday’s Law:
∮E⃗·dl⃗ = -dΦ_B/dt
Meaning: Changing magnetic flux creates electric fields (induction).
4. Ampère-Maxwell Law:
∮B⃗·dl⃗ = μ₀(I + I_d) = μ₀(I + ε₀dΦ_E/dt)
Meaning: Electric currents and changing electric fields create magnetic fields.
Laws 3 and 4 show the beautiful symmetry: changing magnetic fields create electric fields, and changing electric fields create magnetic fields. This mutual creation enables EM wave propagation!
Accelerating charges produce electromagnetic waves. Any charge that changes its velocity (speed or direction) creates EM radiation.
Step 1: Accelerating charge creates changing electric field
As charge accelerates, its electric field pattern changes with time
Step 2: Changing electric field creates magnetic field
By Maxwell’s displacement current, ∂E⃗/∂t creates B⃗
Step 3: This magnetic field is also changing
Because the source (changing E⃗) is changing
Step 4: Changing magnetic field creates electric field
By Faraday’s law, ∂B⃗/∂t creates E⃗
Step 5: Self-perpetuating wave is born!
The wave detaches from source and propagates independently
All EM waves are identical in nature – they differ only in frequency/wavelength. Different names reflect:
Universal Relation: \[c = f\lambda = 3 \times 10^8 \text{ m/s}\]
Energy per photon: \[E = hf = \frac{hc}{\lambda}\]
Higher frequency → Higher energy per photon → More penetrating → Can cause more damage to matter
This explains why gamma rays are dangerous while radio waves are safe!
1. Low Attenuation: Large wavelengths interact weakly with atmospheric molecules and particles
2. Diffraction: Long wavelengths can bend around obstacles (buildings, mountains)
3. Ionospheric Reflection: Certain frequencies reflect off ionosphere, enabling long-distance communication
4. Low Energy: Safe for biological tissues, no ionization
Resonance with Water Molecules:
Water (H₂O) is a polar molecule with natural rotation frequency ~2.45 GHz. Microwave ovens use exactly this frequency!
Heating Mechanism:
Thermal Motion Connection:
All matter above absolute zero has thermal energy. This energy manifests as:
Why This Produces IR:
These thermal motions involve accelerating charges (electrons in atoms/molecules), which emit EM radiation in the infrared range.
Peak wavelength: λ_max = 2.898×10⁻³/T (meters)
Total power radiated: P = σAT⁴ (Stefan-Boltzmann)
Examples:
1. Solar Spectrum Peak: Sun’s blackbody radiation peaks in visible range (~500 nm)
2. Atmospheric Transparency: Atmosphere is most transparent to visible light
3. Water Transparency: Ocean water transmits visible light well
4. Molecular Size Matching: Wavelength comparable to biological molecules enables specific interactions
5. Energy Balance: High enough energy to trigger photochemical reactions, low enough to avoid tissue damage
Why Different Colors?
Different wavelengths have different energies (E = hc/λ). Our eyes have three types of cone cells sensitive to different wavelength ranges, creating color perception.
| Color | λ (nm) | Energy (eV) |
|---|---|---|
| Violet | 400-450 | 3.1-2.8 |
| Blue | 450-500 | 2.8-2.5 |
| Green | 500-575 | 2.5-2.2 |
| Yellow | 575-590 | 2.2-2.1 |
| Red | 620-700 | 2.0-1.8 |
Higher Photon Energy:
UV photons (3-10 eV) have enough energy to break chemical bonds in biological molecules (typically 1-5 eV per bond)
DNA Damage Mechanism:
UV-A (315-400 nm):
UV-B (280-315 nm):
UV-C (100-280 nm):
O₃ + UV → O₂ + O (ozone photolysis absorbs harmful UV-C and most UV-B, protecting life on Earth)
Method 1 – Bremsstrahlung (Braking Radiation):
Method 2 – Characteristic X-rays:
High Photon Energy (1-100 keV):
Small Wavelength (0.01-10 nm):
Different tissues absorb X-rays differently:
This creates contrast in X-ray images!
Nuclear Transitions:
Radioactive Decay:
Often accompanies alpha or beta decay when daughter nucleus is in excited state
Extremely High Energy:
High Penetration:
1. Absorption:
2. Scattering:
3. Transmission:
Maxwell’s relation in matter:
v = 1/√(μ₀μᵣε₀εᵣ) = c/√(μᵣεᵣ)
For non-magnetic materials (μᵣ ≈ 1):
n = c/v = √εᵣ
Physical meaning of εᵣ:
Relative permittivity εᵣ > 1 means material can store more electric field energy than vacuum, slowing down EM waves
\[n = \frac{c}{v} = \sqrt{\epsilon_r \mu_r} \approx \sqrt{\epsilon_r}\]
\[v = \frac{c}{n}\] (speed in medium)
Definition: Polarization describes the direction of oscillation of the electric field vector in an EM wave.
Why Only Electric Field?
1. Linear Polarization:
E⃗ oscillates in fixed direction: E⃗ = E₀cos(kz-ωt)x̂
2. Circular Polarization:
Ex = E₀cos(kz-ωt), Ey = E₀cos(kz-ωt±π/2)
E⃗ vector rotates, maintaining constant magnitude
3. Elliptical Polarization:
Ex = E₁cos(kz-ωt), Ey = E₂cos(kz-ωt+δ)
E⃗ vector traces ellipse
When linearly polarized light hits polarizer at angle θ:
Only component E⃗cosθ parallel to polarizer passes through
Intensity ∝ E², so I = I₀cos²θ
Maximum transmission: θ = 0° (parallel)
Zero transmission: θ = 90° (perpendicular)
(a) Wavelength:
λ = c/f = (3×10⁸)/(5×10¹⁴) = 6×10⁻⁷ m = 600 nm
This is orange light!
(b) Magnetic field amplitude:
For EM waves: E₀/B₀ = c
B₀ = E₀/c = 100/(3×10⁸) = 3.33×10⁻⁷ T
(c) Average energy density:
⟨u⟩ = ε₀E₀²/2 = (8.854×10⁻¹² × 100²)/2
= 4.427×10⁻⁸ J/m³
(d) Intensity:
I = cε₀E₀²/2 = (3×10⁸ × 8.854×10⁻¹² × 100²)/2
= 1.328×10¹ = 13.28 W/m²
(e) Radiation pressure:
P = I/c = 13.28/(3×10⁸) = 4.43×10⁻⁸ Pa
Key Concepts:
Step-by-step Solution:
After 1st polarizer (0°):
Unpolarized light → linearly polarized
I₁ = I₀/2 = 100/2 = 50 W/m²
(½ factor for unpolarized → polarized conversion)
After 2nd polarizer (45° to 1st):
I₂ = I₁cos²(45°) = 50 × (1/√2)² = 50 × (1/2) = 25 W/m²
After 3rd polarizer (90° to 2nd, i.e., 45° total):
I₃ = I₂cos²(45°) = 25 × (1/2) = 12.5 W/m²
Final Answer: 50, 25, 12.5 W/m²
1. Ground Wave (Surface Wave):
2. Sky Wave (Ionospheric):
3. Space Wave (Line-of-Sight):
Line-of-Sight Distance:
\[d = \sqrt{2Rh}\] (for antenna height h)
Where R = Earth’s radius ≈ 6400 km
No Medium Required:
Unlike sound waves, EM waves don’t need medium, so there’s no “wind effect.” Only relative motion between source and observer matters.
Relativistic Effects:
At high speeds (v ≈ c), time dilation becomes important
Source Moving Toward Observer:
In time T, source emits N waves at frequency f
Distance traveled by source = vT
Waves compressed into length = cT – vT
Observed wavelength λ’ = (cT – vT)/N = (c – v)/f
Observed frequency f’ = c/λ’ = cf/(c – v) ≈ f(1 + v/c)
Source Moving Away:
f’ ≈ f(1 – v/c)
General Doppler Formula:
\[f’ = f\sqrt{\frac{c + v_r}{c + v_s}}\]
For v << c:
\[f’ \approx f\left(1 + \frac{v}{c}\right)\] (approaching)
Step 1: Identify the Type
Step 2: List Known/Unknown
Step 3: Choose Key Relations
Step 4: Check Dimensions
Always verify units match expected result
Step 5: Physical Reasonableness
Does answer make physical sense?
1. Frequency vs Wavelength Confusion
2. Energy Density Calculation
3. Polarization Direction
4. Speed of Light Formula
5. Displacement Current