🌊 Wave Properties & Representation

Master wave calculations and precision in EM wave measurements

Precision in Wave Calculations

In electromagnetic waves, precision matters. A small error in frequency calculation translates to wrong wavelength, wrong energy, and wrong classification in the spectrum.

Most calculation errors come from:

Using wrong units (Hz vs MHz, m vs nm)
Forgetting to convert wavelength to meters
Mixing up c and v (vacuum vs medium)

📏 Wavelength (λ)

Distance between two consecutive peaks or troughs.

λ = c/ν (in vacuum)

λ = v/ν (in medium)

Common Units:

1 m = 10⁹ nm (nanometer)
1 m = 10¹⁰ Å (angstrom)
1 nm = 10 Å

For visible light: λ ≈ 400-700 nm
For radio waves: λ can be kilometers
For X-rays: λ ≈ 0.01-10 nm

📡 Frequency (ν)

Number of oscillations per second.

ν = c/λ (in vacuum)

ν = v/λ (in medium)

Common Units:

1 Hz = 1 oscillation/second
1 kHz = 10³ Hz
1 MHz = 10⁶ Hz
1 GHz = 10⁹ Hz
1 THz = 10¹² Hz

Frequency remains constant when wave enters a medium. Only speed and wavelength change!

Wave Behavior in Different Media

Vacuum → Medium Transition ▼

When EM wave goes from vacuum to medium:

Property	Change	Reason
Frequency (ν)	Unchanged	Source determines frequency
Speed (v)	Decreases	v = c/n, where n > 1
Wavelength (λ)	Decreases	Since v↓ and ν unchanged, λ = v/ν ↓

λ_medium = λ_vacuum / n

Refractive Index & Speed ▼

n = c/v = √(μ_r ε_r)

where μ_r and ε_r are relative permeability and permittivity.

For most non-magnetic materials:

n ≈ √ε_r

JEE loves to ask: "If wavelength in medium is X nm, what's wavelength in vacuum?"
Answer: λ_vacuum = n × λ_medium

Phase & Phase Difference ▼

Phase of wave:

φ = kx - ωt

Phase difference between two points:

Δφ = k·Δx = (2π/λ)·Δx

Path difference and phase difference:

Δφ = (2π/λ) × (path difference)

If path difference = λ, phase difference = 2π
If path difference = λ/2, phase difference = π
If path difference = λ/4, phase difference = π/2

Worked Examples: Precision Calculations

Example 1: Wavelength Calculation

Given: A radio station broadcasts at 95.5 MHz. Find wavelength.

Step 1: Convert frequency to Hz

ν = 95.5 MHz = 95.5 × 10⁶ Hz

Step 2: Use c = νλ

λ = c/ν = (3 × 10⁸)/(95.5 × 10⁶)

Step 3: Calculate

λ = 3.14 m

This is in the radio wave range. Wavelength is about 3 meters—that's why radio antennas are often a few meters long!

Example 2: Medium Transition

Given: Yellow light (λ = 600 nm in vacuum) enters glass (n = 1.5). Find wavelength and frequency in glass.

Step 1: Frequency in vacuum

ν = c/λ = (3×10⁸)/(600×10⁻⁹) = 5×10¹⁴ Hz

Step 2: Frequency in glass

ν_glass = ν_vacuum = 5×10¹⁴ Hz (unchanged!)

Step 3: Wavelength in glass

λ_glass = λ_vacuum / n = 600/1.5 = 400 nm

Color depends on frequency, not wavelength. So light remains yellow in glass even though wavelength changed!

Example 3: E and B Fields

Given: In an EM wave, E₀ = 900 V/m. Find B₀.

Use: E₀/B₀ = c

B₀ = E₀/c = 900/(3×10⁸)

Calculate:

B₀ = 3 × 10⁻⁶ T = 3 μT

Common error: Forgetting that B is much smaller numerically than E. Always verify your answer makes sense!

Precision Checklist for Exam

Before Calculation:

✓ Identify if wave is in vacuum or medium
✓ Check all units (convert if needed)
✓ Note what's constant (usually ν)
✓ Note what changes (v and λ in medium)

During Calculation:

✓ Use c = 3×10⁸ m/s (vacuum)
✓ Use v = c/n (medium)
✓ Keep significant figures consistent
✓ Write intermediate steps clearly

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