A beam contains infrared light of a single wavelength, 1000 nm, and monochromatic UV at 100 nm, both of the same intensity Are there more 100 nm photons or more 1000 nm photons ?

Explanation

To determine whether there are more 100 nm photons (UV) or 1000 nm photons (infrared) in a beam containing both, we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy of a photon is inversely proportional to its wavelength. This relationship can be expressed using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( c \) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \( \lambda \) is the wavelength of the photon.

### Step 2: Calculate the energy of the photons
1. For the UV photon (100 nm):
\[ E_{UV} = \frac{hc}{\lambda_{UV}} = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{100 \times 10^{-9} \, \text{m}} \]
\[ E_{UV} = \frac{(6.626 \times 3)}{100} \times 10^{-19} \approx 1.9878 \times 10^{-18} \, \text{J} \]

2. For the infrared photon (1000 nm):
\[ E_{IR} = \frac{hc}{\lambda_{IR}} = \frac{(6.626 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{1000 \times 10^{-9} \, \text{m}} \]
\[ E_{IR} = \frac{(6.626 \times 3)}{1000} \times 10^{-19} \approx 1.9878 \times 10^{-21} \, \text{J} \]

### Step 3: Relate intensity to the number of photons
The intensity \( I \) of the beam is related to the number of photons \( n \) and the energy of each photon \( E \):
\[ I = n \cdot E \]
From this, we can express the number of photons:
\[ n = \frac{I}{E} \]

### Step 4: Calculate the number of photons for each wavelength
1. For the UV photons:
\[ n_{UV} = \frac{I}{E_{UV}} \]

2. For the infrared photons:
\[ n_{IR} = \frac{I}{E_{IR}} \]

### Step 5: Compare the number of photons
Since both beams have the same intensity \( I \):
\[ \frac{n_{UV}}{n_{IR}} = \frac{E_{IR}}{E_{UV}} \]
Since \( E_{UV} > E_{IR} \), it follows that:
\[ \frac{n_{UV}}{n_{IR}} < 1 \Rightarrow n_{UV} < n_{IR} \]

### Conclusion
Thus, there are more 1000 nm photons (infrared) than 100 nm photons (UV) in the beam.

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