How many photons with wavelength 300 nm (nanometers) does it take to equal the
energy from one photon of 100 nm?

To compare the energy of photons with different wavelengths, we can use the formula for the energy of a photon:

\[
E = \frac{h c}{\lambda}
\]

where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J·s}) \),
- \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \),
- \( \lambda \) is the wavelength of the photon.

Now, let's calculate the energy of a photon with a wavelength of 100 nm and 300 nm.

### Step 1: Convert the wavelengths into meters:
- For \( 100 \, \text{nm} \): \( \lambda_1 = 100 \times 10^{-9} \, \text{m} \)
- For \( 300 \, \text{nm} \): \( \lambda_2 = 300 \times 10^{-9} \, \text{m} \)

### Step 2: Calculate the energy of each photon:
1. **Energy of a 100 nm photon**:
   \[
   E_1 = \frac{h c}{\lambda_1} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{100 \times 10^{-9}} = 1.9878 \times 10^{-18} \, \text{J}
   \]
   
2. **Energy of a 300 nm photon**:
   \[
   E_2 = \frac{h c}{\lambda_2} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{300 \times 10^{-9}} = 6.626 \times 10^{-19} \, \text{J}
   \]

### Step 3: Determine how many 300 nm photons are required to equal the energy of one 100 nm photon:
\[
\text{Number of photons} = \frac{E_1}{E_2} = \frac{1.9878 \times 10^{-18}}{6.626 \times 10^{-19}} = 3
\]

So, it takes 3 photons with a wavelength of 300 nm to equal the energy of one photon with a wavelength of 100 nm.

Would you like more details on any of these steps?

### Related Questions:
1. How does wavelength affect the energy of a photon?
2. What is the relationship between photon energy and frequency?
3. How do you convert between wavelength and frequency in the context of photon energy?
4. What is Planck’s constant and how is it derived?
5. How does the speed of light factor into photon energy calculations?

### Tip:  
The shorter the wavelength, the higher the energy of a photon since energy is inversely proportional to wavelength!

How many photons with wavelength 300 nm (nanometers) does it take to equal the energy from one photon of 100 nm?

Understand how many photons of 300 nm wavelength are required to match the energy of one 100 nm photon. This problem explores key concepts in quantum mechanics, including Planck's constant, the speed of light, and the inverse relationship between energy and wavelength. Suitable for advanced high school or undergraduate physics students.

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