#wave number

Applications of EM Waves in Communication and Medicine

An electromagnetic wave has a wave number \( k = 1 \, \text{rad/m} \). What is its wavelength in vacuum?

**Microwaves in ovens** cause water molecules to rotate at 2.45 GHz, friction heats food, penetration depth few cm, efficient heating, also radar uses reflection of microwaves from objects, Doppler shift measures speed, medical diathermy uses microwaves for tissue heating. The wave number k = (2 π/λ) . Given k = 1 rad/m , we have λ = (2 π/k) = (2 π/1) ≈ 6.28 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 6.28 m, illustrating EM wave transverse nature and Maxwell's displacement current concept.

Ref: NCERT > Physics Book > Electromagnetic Waves > Applications of EM Waves in Communication and Medicine

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EM Wave Characteristics - Transverse Nature and E/B Ratio

An electromagnetic wave has a wave number \( k = 5 \, \text{rad/m} \). What is its wavelength in vacuum?

**Transverse nature** means E and B perpendicular to direction, e.g., wave propagating along z, E along x, B along y, Poynting vector S = E×B/μ₀ along z, energy flow direction. E and B in phase, maxima together, ratio fixed c. The wave number k = (2 π/λ) . Given k = 5 rad/m , we have λ = (2 π/k) = (2 π/5) ≈ 1.26 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 1.26 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 has a wave number \( k = 2 \, \text{rad/m} \). What is its wavelength in vacuum?

**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. The wave number k = (2 π/λ) . Given k = 2 rad/m , we have λ = (2 π/k) = (2 π/2) = π ≈ 3.14 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 3.14 m, 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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Ultraviolet, X-rays, Gamma Rays and Their Properties

An electromagnetic wave has a wavelength of \( 3 \, \text{m} \) in vacuum. What is its wave number \( k \)?

**X-rays in medicine** treat certain cancers because high-energy photons ionize and damage DNA of malignant cells, also imaging because bones absorb more than tissue, contrast. Gamma rays from Co-60 used for radiotherapy, energy ≈1.17 and 1.33 MeV, highly penetrating. The wave number k = (2 π/λ) . Given λ = 3 m , k = (2 π/3) ≈ 2.09 rad/m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 2.09 rad/m, 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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Ultraviolet, X-rays, Gamma Rays and Their Properties

An electromagnetic wave has a wave number \( k = 4 \, \text{rad/m} \). What is its wavelength in vacuum?

**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. The wave number k = (2 π/λ) . Given k = 4 rad/m , we have λ = (2 π/k) = (2 π/4) ≈ 1.57 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 1.57 m, 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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Ultraviolet, X-rays, Gamma Rays and Their Properties

An electromagnetic wave has a wave number \( k = 0.5 \, \text{rad/m} \). What is its wavelength in vacuum?

**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. The wave number k = (2 π/λ) . Given k = 0.5 rad/m , we have λ = (2 π/k) = (2 π/0.5) ≈ 12.56 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 12.56 m, 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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Ultraviolet, X-rays, Gamma Rays and Their Properties

An electromagnetic wave has a wave number \( k = 3 \, \text{rad/m} \). What is its wavelength in vacuum?

**Ultraviolet** λ≈10-400 nm, produced by atomic transitions and Sun, causes fluorescence, sterilization, harmful to eyes/skin, absorbed by ozone layer. **X-rays** λ≈10⁻² to 10 nm, produced by rapid deceleration of energetic electrons in X-ray tube, used in medicine for diagnosis (shadow of bones) and cancer therapy due to high ionization and penetration, detected by photographic film and Geiger counter. The wave number k = (2 π/λ) . Given k = 3 rad/m , we have λ = (2 π/k) = (2 π/3) ≈ 2.09 m . Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields 2.09

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

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Transverse and Longitudinal Waves

What is the physical significance of the wave number in a progressive wave?

**Wave classification** depends on particle vibration relative to propagation. Longitudinal waves have particle oscillation parallel to propagation, creating compressions and rarefactions as in sound in air; transverse have perpendicular oscillation. Tuning fork generates longitudinal sound because air cannot sustain shear. Wave number k = (2π/λ) represents the number of radians per unit distance, indicating how many wave cycles fit into a unit length, inversely related to wavelength. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Spatial fre

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Wave Equation and Displacement Relation

A wave is described by \( y(x, t) = 0.04 \sin (20x - 50t) \), where \( x \) and \( y \) are in meters and \( t \) in sec

**Wave equation** y(x,t) = A sin(kx - ωt + φ) describes displacement of progressive harmonic wave, where k = 2π/λ wave number (rad/m), ω = 2πf angular frequency (rad/s), v = ω/k wave speed (m/s). Sign of ωt indicates direction, amplitude A is maximum displacement. Compare with y = a sin (kx - ω t) , k = 20 rad/m . Wavelength: λ = (2π/k) = (2π/20) = 0.314 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0.314 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Equation and Displacement Relation

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