Question

The number of photons emitted by a monochromatic (single frequency) infrared range finder of power 1 mW and wavelength of 1000 nm, in 0.1 second is \(x \times 10^{13}\). The value of x is _________. (Nearest integer)
(h = 6.63 \(\times\) 10\(^{-34}\) Js, c=3.00 \(\times\) 10\(^8\) ms\(^{-1}\))

Hint:

The number of photons emitted per second is simply the power of the source divided by the energy of one photon (\(N/t = P/E_p\)). You can calculate this rate first and then multiply by the time, or calculate the total energy first as done here. Both methods are equivalent.

Answer 1

Correct Answer: 50

Step 1: Understanding the Question:
We are given the power, wavelength, and time duration of an infrared source. We need to calculate the total number of photons emitted in that time.
Step 2: Calculate the Energy of a Single Photon:
The energy of a single photon (E\(_p\)) is given by the Planck-Einstein relation:
\[ E_p = hf = \frac{hc}{\lambda} \] Given:
- \(h = 6.63 \times 10^{-34}\) Js
- \(c = 3.00 \times 10^8\) m/s
- \(\lambda = 1000\) nm = \(1000 \times 10^{-9}\) m = \(10^{-6}\) m
\[ E_p = \frac{(6.63 \times 10^{-34} \, \text{Js}) \times (3.00 \times 10^8 \, \text{m/s})}{10^{-6} \, \text{m}} = 1.989 \times 10^{-19} \, \text{J} \] Step 3: Calculate the Total Energy Emitted:
The total energy (E\(_t\)) emitted by the source is its power (P) multiplied by the time (t).
Given:
- \(P = 1\) mW = \(1 \times 10^{-3}\) W = \(1 \times 10^{-3}\) J/s
- \(t = 0.1\) s
\[ E_t = P \times t = (1 \times 10^{-3} \, \text{J/s}) \times (0.1 \, \text{s}) = 1 \times 10^{-4} \, \text{J} \] Step 4: Calculate the Number of Photons:
The total number of photons (N) is the total energy emitted divided by the energy of a single photon.
\[ N = \frac{E_t}{E_p} = \frac{1 \times 10^{-4} \, \text{J}}{1.989 \times 10^{-19} \, \text{J}} \approx 0.5027 \times 10^{15} = 5.027 \times 10^{14} \] Step 5: Find the value of x:
We are given that the number of photons is \(x \times 10^{13}\).
\[ 5.027 \times 10^{14} = x \times 10^{13} \] \[ 50.27 \times 10^{13} = x \times 10^{13} \] \[ x = 50.27 \] The value of x to the nearest integer is 50.