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Question: A parallel beam of uniform monochromatic light of wavelength 546 nm has an intensity of 200 W/m2. Th...

A parallel beam of uniform monochromatic light of wavelength 546 nm has an intensity of 200 W/m2. The number of photons in 1 mm3 of this radiation is:

Answer

1830

Explanation

Solution

The intensity of a parallel beam of light is given by I=ucI = u c, where uu is the energy density of the radiation and cc is the speed of light.

The energy density uu is related to the number of photons per unit volume nn and the energy of a single photon EE by u=nEu = nE.

The energy of a single photon with wavelength λ\lambda is E=hcλE = \frac{hc}{\lambda}, where hh is Planck's constant and cc is the speed of light.

Combining these equations, we get: I=(nE)c=n(hcλ)c=nhc2λI = (nE)c = n \left(\frac{hc}{\lambda}\right) c = n \frac{hc^2}{\lambda}

The number of photons per unit volume is n=Iλhc2n = \frac{I \lambda}{hc^2}. This gives the number of photons per cubic meter (m⁻³).

We are given:

  • Intensity, I=200 W/m2I = 200 \text{ W/m}²
  • Wavelength, λ=546 nm=546×10−9 m\lambda = 546 \text{ nm} = 546 \times 10⁻⁹ \text{ m}
  • Volume, V=1 mm3=(10−3 m)3=10−9 m3V = 1 \text{ mm}³ = (10⁻³ \text{ m})³ = 10⁻⁹ \text{ m}³
  • Planck's constant, h≈6.63×10−34 J sh \approx 6.63 \times 10⁻³⁴ \text{ J s}
  • Speed of light, c≈3×108 m/sc \approx 3 \times 10⁸ \text{ m/s}

The number of photons in the volume VV is N=n×VN = n \times V. N=Iλhc2×VN = \frac{I \lambda}{hc^2} \times V

Substitute the given values: N=(200 W/m2)×(546×10−9 m)(6.63×10−34 J s)×(3×108 m/s)2×(10−9 m3)N = \frac{(200 \text{ W/m}²) \times (546 \times 10⁻⁹ \text{ m})}{(6.63 \times 10⁻³⁴ \text{ J s}) \times (3 \times 10⁸ \text{ m/s})²} \times (10⁻⁹ \text{ m}³) N=200×546×10−96.63×10−34×(9×1016)×10−9N = \frac{200 \times 546 \times 10⁻⁹}{6.63 \times 10⁻³⁴ \times (9 \times 10¹⁶)} \times 10⁻⁹ N=200×546×10−959.67×10−18×10−9N = \frac{200 \times 546 \times 10⁻⁹}{59.67 \times 10⁻¹⁸} \times 10⁻⁹ N=109200×10−959.67×10−18×10−9N = \frac{109200 \times 10⁻⁹}{59.67 \times 10⁻¹⁸} \times 10⁻⁹ N = \frac{1.092 \times 10⁵ \times 10⁻⁹}}{5.967 \times 10¹ \times 10⁻¹⁸}} \times 10⁻⁹ N = \frac{1.092 \times 10⁻⁴}}{5.967 \times 10⁻¹⁷}} \times 10⁻⁹ N=1.0925.967×10(−4−(−17))×10−9N = \frac{1.092}{5.967} \times 10^{(⁻⁴ - (⁻¹⁷))} \times 10⁻⁹ N=1.0925.967×10(−4+17)×10−9N = \frac{1.092}{5.967} \times 10^{(⁻⁴ + 17)} \times 10⁻⁹ N=1.0925.967×1013×10−9N = \frac{1.092}{5.967} \times 10¹³ \times 10⁻⁹ N=1.0925.967×104N = \frac{1.092}{5.967} \times 10⁴ N≈0.1830×104N \approx 0.1830 \times 10⁴ N≈1830N \approx 1830

Rounding to the nearest integer, the number of photons is 1830.