Question

Find number of photons emitted per second by a light source of wavelength \(663\,\text{nm}\) at power \(6\,\text{mW}\). If \(h = 6.63 \times 10^{-34}\,\text{Js}\) is \(N \times 10^{15}\), then find \(N\).

Hint:

Photon count problems are easiest when you remember: {Number of photons per second = Power ÷ Energy of one photon}.

Answer 1

Correct Answer: 20

To find the number of photons emitted per second by a light source, we first need to determine the energy of a single photon. The energy \(E\) of a photon is given by:

\(E = \frac{hc}{\lambda}\),

where \(h\) is Planck's constant, \(c\) is the speed of light in vacuum, and \(\lambda\) is the wavelength.

Given: \(h = 6.63 \times 10^{-34}\,\text{Js}\), \(c = 3 \times 10^8\,\text{m/s}\), and \(\lambda = 663\,\text{nm} = 663 \times 10^{-9}\,\text{m}\).

Substitute these values:

\(E = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{663 \times 10^{-9}}\)

\(= \frac{1.989 \times 10^{-25}}{663 \times 10^{-9}}\)

\(= 2.997 \times 10^{-19}\,\text{J/photon}\).

Next, calculate the number of photons emitted per second by dividing the total power by the energy of one photon. The total power \(P\) is \(6\,\text{mW} = 6 \times 10^{-3}\,\text{W}\).

Number of photons per second \(n\) is given by:

\(n = \frac{P}{E}\)

\(= \frac{6 \times 10^{-3}}{2.997 \times 10^{-19}}\)

\(= 2.002 \times 10^{16}\,\text{photons/second}\).

If \(n = N \times 10^{15}\), then:

\(2.002 \times 10^{16} = N \times 10^{15}\)

\(N = \frac{2.002 \times 10^{16}}{10^{15}}\)

\(= 20.02\).

Thus, \(N\) is approximately 20, which falls within the range (20, 20). Hence, \(N = 20\).