#wave calculation

Production of EM Waves and Hertz Experiment

An electromagnetic wave has a magnetic field given by \( B_y = 5 \times 10^{-8} \sin(3 \times 10^3 z - 9 \times 10^{11}

**Aerials produce radio waves** by rapid acceleration/deceleration of electrons in antenna driven by AC, frequency equals driving frequency, for 60 MHz, λ=c/f=3×10⁸/60×10⁶=5 m, half-wave antenna length λ/2=2.5 m, efficient radiation when antenna size comparable to λ. Comparing with B_y = B₀ sin(kz - ω t) , we have k = 3 × 10³ rad/m . Wavelength λ = (2 π/k) = (2 π/3 × 10³) ≈ 2.09 × 10⁻³ m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 2.09 × 10⁻³ m, illustrating EM wave transverse nature and Maxwell's displacement current concept.

Ref: NCERT > Physics Book > Electromagnetic Waves > Production of EM Waves and Hertz Experiment

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EM Wave Characteristics - Transverse Nature and E/B Ratio

An electromagnetic wave in vacuum has a magnetic field amplitude of \( B_0 = 4 \times 10^{-8} \, \text{T} \). What is th

**EM wave in vacuum** transverse, E and B perpendicular to propagation and to each other, E×B along propagation, in phase, E/B = c =3×10⁸ m/s, c =1/√(μ₀ ε₀), μ₀=4π×10⁻⁷ H/m, ε₀=8.85×10⁻¹² F/m. For E₀=45 V/m, B₀=E₀/c=45/3×10⁸=1.5×10⁻⁷ T=150 nT, illustrating B much smaller than E. Using E₀ = B₀ c , we have E₀ = (4 × 10⁻⁸) × (3 × 10⁸) = 12 V/m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 12 V/m, illustrating EM wave transverse nature and Maxwell's displacement current concept.

Ref: NCERT > Physics Book > Electromagnetic Waves > EM Wave Characteristics - Transverse Nature and E/B Ratio

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Ultraviolet, X-rays, Gamma Rays and Their Properties

An electromagnetic wave in vacuum has a wavelength of \( 12 \, \text{m} \). What is its frequency? (Given \( c = 3 \time

**Gamma rays** λ10¹⁹ Hz, produced by nuclear transitions and radioactive decay, associated with nuclear processes because nuclear energy levels MeV vs atomic eV, highest photon energy, used in cancer treatment due to high penetration and cell damage, also sterilization, astronomy. Using v λ = c , we have v = (c/λ) = (3 × 10⁸/12) = 2.5 × 10⁷ Hz . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 2.5 × 10⁷ Hz, illustrating EM wave transverse nature and Maxwell's displacement current concept.

Ref: NCERT > Physics Book > Electromagnetic Waves > Ultraviolet, X-rays, Gamma Rays and Their Properties

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Doppler Effect

A string fixed at both ends has a length of 1.6 m and a fundamental frequency of 62.5 Hz. What is the speed of the wave?

**Relative motion** changes effective wavelength encountered. For moving source approaching stationary observer, wavelength ahead λ' = (v - v_s)/f, so f' = v/λ' = f·v/(v - v_s) > f, basis for calculating apparent pitch shift in sound. Fundamental: v₁ = (v/2L) . 62.5 = (v/2 × 1.6) ⇒ v = 62.5 × 3.2 = 200 m/s . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 200 m/s, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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