Question

a 75 w lamp emits light of wavelength 460 nm uniformly in all directions. what is the photon flux (photons per unit area per unit time) on a small screen at a distance 1.8 m from the lamp?

8.5 × 10¹⁸ photons/m²·s
1.7 × 10¹⁹ photons/m²·s
7.7 × 10¹⁸ photons/m²·s
5.4 × 10¹⁹ photons/m²·s
4.3 × 10¹⁸ photons/m²·s

Explanation:

Step1: Recall photon energy formula

The energy of a single photon is given by \( E = \frac{hc}{\lambda} \), where \( h = 6.626\times 10^{-34}\, \text{J·s} \) (Planck's constant), \( c = 3\times 10^{8}\, \text{m/s} \) (speed of light), and \( \lambda \) is the wavelength. First, convert \( \lambda = 460\, \text{nm}=460\times 10^{-9}\, \text{m} \).

Calculate \( E \):

$$ E=\frac{6.626\times 10^{-34}\times 3\times 10^{8}}{460\times 10^{-9}}=\frac{1.9878\times 10^{-25}}{4.6\times 10^{-7}}\approx 4.32\times 10^{-19}\, \text{J} $$

Step2: Find total number of photons per second

The power of the lamp \( P = 75\, \text{W}=75\, \text{J/s} \). The number of photons per second \( n \) is given by \( n=\frac{P}{E} \).

Substitute values:

$$ n = \frac{75}{4.32\times 10^{-19}}\approx 1.736\times 10^{20}\, \text{photons/s} $$

Step3: Calculate photon flux

The lamp emits uniformly in all directions, so the light is spread over a sphere of radius \( r = 1.8\, \text{m} \). The surface area of a sphere is \( A = 4\pi r^{2} \).

Calculate \( A \):

$$ A = 4\pi(1.8)^{2}=4\pi\times 3.24\approx 40.715\, \text{m}^2 $$

Photon flux \( \Phi=\frac{n}{A} \). Substitute \( n \) and \( A \):

$$ \Phi=\frac{1.736\times 10^{20}}{40.715}\approx 4.26\times 10^{18}\, \text{photons/m}^2\text{·s} $$

(Close to \( 4.3\times 10^{18}\, \text{photons/m}^2\text{·s} \) due to rounding differences in steps)

Answer:

4.3 × 10¹⁸ photons/m²·s (the last option: 4.3 × 10¹⁸ photons/m²·s)