Chapter 20: Electromagnetic Waves (Numerical Problems)

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Chapter Overview & SLOs

How are wavelength and frequency calculated for electromagnetic waves? All electromagnetic waves travel through a vacuum at the constant speed of light, c ≈ 3.0 × 10⁸ m/s. They adhere strictly to the fundamental wave equation c = λf.

Rearranged formulas:

• λ = c/f (wavelength = speed of light / frequency)
• f = c/λ (frequency = speed of light / wavelength)

Example 1 - Wavelength calculation: A wave with frequency f = 5.0 × 10¹⁴ Hz: λ = c/f = (3.0 × 10⁸)/(5.0 × 10¹⁴) = 6.0 × 10⁻⁷ m = 600 nm.

How do we determine the energy carried by individual photons? According to quantum mechanics, electromagnetic radiation travels as discrete packets of energy called photons. The energy of a single photon is directly proportional to its frequency, computed via Planck's equation E = hf, where h = 6.626 × 10⁻³⁴ J·s.

Example 2 - Gamma-ray photon energy: For a gamma-ray photon with frequency f = 2.0 × 10²⁰ Hz: E = hf = (6.626 × 10⁻³⁴)(2.0 × 10²⁰) = 1.33 × 10⁻¹³ J.

How do we determine the linear momentum carried by photons? The linear momentum carried by a photon is inversely proportional to its wavelength: p = h/λ.

Example 3 - Photon momentum: For shortest visible light wavelength λ = 380 nm = 380 × 10⁻⁹ m: p = h/λ = (6.626 × 10⁻³⁴)/(380 × 10⁻⁹) = 1.74 × 10⁻²⁷ kg·m/s.

How do we convert photon energy from Joules to electron-volts (eV)? Divide the energy in Joules by 1.6 × 10⁻¹⁹ J/eV.

Example 4 - Energy in eV: For λ = 380 nm, first calculate E = hc/λ = (6.626 × 10⁻³⁴ × 3.0 × 10⁸)/(380 × 10⁻⁹) = 5.23 × 10⁻⁹ J. Then convert to eV: E = (5.23 × 10⁻¹⁹)/(1.6 × 10⁻¹⁹) ≈ 3.27 eV.

Wavelength conversions: To convert meters to nanometers, multiply by 10⁹ nm/m. Example: 6.0 × 10⁻⁷ m = 600 nm.

Spectral classification: A calculated wavelength of 265 nm or frequency of 1.136 × 10¹⁵ Hz points to the ultraviolet (UV) spectrum.

Important note for scientific notation: When solving E = hf, use brackets to group Planck's constant: (6.626 × 10⁻³⁴) × frequency to prevent order-of-operation errors. The speed of light c = 3.0 × 10⁸ m/s is an essential constant to memorize.

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we apply the fundamental wave speed equation $c = \lambda f$ to solve numerical problems? Students will manipulate expressions to calculate unknown wavelengths or frequencies across various bands, converting raw metrics into nanometers (nm) for accurate spectral classification.
• How do we evaluate the energetic profile of quantum radiation using Planck's relation $E = hf$? Students will systematically determine photon energy values for high-frequency bands, such as calculating how a gamma-ray photon at $2.0 \times 10^{20}\text{ Hz}$ outputs a value of $1.33 \times 10^{-13}\text{ J}$.
• How do we compute the linear momentum of light particles across distinct spectral boundaries using $p = h/\lambda$? Students will resolve simultaneous equations matching energy and momentum values for the shortest visible light thresholds down to $380\text{ nm}$.
• How do we correlate calculated numerical attributes back to specific diagnostic domains? Students will deduce whether a calculated wavelength of $265\text{ nm}$ or frequency of $1.136 \times 10^{15}\text{ Hz}$ points to ultraviolet, infrared, or microwave segments within a standard reference system.

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