Question 4 of 10intermediate🔧 ApplyNumerical5 marks

Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW. (a) Find the energy and momentum of each photon in the light beam, (b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area), and (c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?

(a)
Answer

Given: Wavelength λ = 632.8 nm = 632.8×10632.8 \times 10⁻⁹ m

Energy of photon: E = hν = hc/λ E = (6.63×106.63 \times 10⁻³⁴ J s × 3×103 \times 10⁸ m/s)/(632.8×10632.8 \times 10⁻⁹ m) E = 3.14×103.14 \times 10⁻¹⁹ J

Momentum of photon: p = hν/c = h/λ p = (6.63×106.63 \times 10⁻³⁴ J s)/(632.8×10632.8 \times 10⁻⁹ m) p = 1.05×101.05 \times 10⁻²⁷ kg m/s

(b)
Answer

Given: Power P = 9.42 mW = 9.42×109.42 \times 10⁻³ W

From the relation P = N × E (where N is number of photons per second):

N = P/E N = (9.42×109.42 \times 10⁻³ W)/(3.14×103.14 \times 10⁻¹⁹ J) N = 3×103 \times 10¹⁶ photons per second

(c)
Answer

Given: Momentum of photon p = 1.05×101.05 \times 10⁻²⁷ kg m/s Mass of hydrogen atom m = 1.67×101.67 \times 10⁻²⁷ kg

For hydrogen atom to have same momentum as photon: p = mv

v = p/m v = (1.05×101.05 \times 10⁻²⁷ kg m/s)/(1.67×101.67 \times 10⁻²⁷ kg) v = 0.63 m/s

Explanation

This question from NCERT Chapter 11 tests understanding of photon properties. Part (a) uses E = hc/λ and p = h/λ from the photon picture of electromagnetic radiation. Part (b) follows the methodology of Example 11.1 where power equals photon flux times energy per photon (P = NE). Part (c) applies momentum conservation by equating photon momentum to atomic momentum (p = mv). The mass of hydrogen atom (1.67×101.67 \times 10⁻²⁷ kg) is a fundamental constant.

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