Question

The ratio of the power of a light source \( S_1 \) to that of the light source \( S_2 \) is 2. \( S_1 \) is emitting \( 2 \times 10^{15} \) photons per second at 600 nm. If the wavelength of the source \( S_2 \) is 300 nm, then the number of photons per second emitted by \( S_2 \) is ________________ \( \times 10^{14} \).

Hint:

The number of photons emitted is inversely proportional to the energy of each photon. For shorter wavelengths, the energy per photon is higher, so fewer photons are emitted for the same power.

Answer 1

Step 1: Understand the Given Information

We are given the following information:

• The ratio of the power of light sources \( S_1 \) and \( S_2 \) is 2, i.e., \( \frac{P_1}{P_2} = 2 \).
  
• Source \( S_1 \) emits \( 2 \times 10^{15} \) photons per second at a wavelength of 600 nm.
  
• The wavelength of source \( S_2 \) is 300 nm.
  

Step 2: Relationship Between Power, Number of Photons, and Energy

The power of a light source is related to the energy emitted per second. The energy of a photon is given by:
E = (6.626 × 10^-34) J·s × (3.0 × 10⁸) m/s / λ

where:
- \( h \) is Planck's constant (6.626 × 10^-34 J·s),
- \( c \) is the speed of light (3.0 × 10⁸ m/s),
- \( λ \) is the wavelength of the light.
The power emitted by the source is the total energy emitted per second, which can be written as:
P = Number of photons × E
Therefore, the number of photons emitted per second is proportional to the power and inversely proportional to the energy per photon.

Step 3: Relating Power and Photons for Both Sources

The ratio of powers \( \frac{P_1}{P_2} = 2 \) implies:
P₁ / P₂ = (N₁ × E₁) / (N₂ × E₂) = 2
where:
- N₁ and N₂ are the number of photons per second for \( S_1 \) and \( S_2 \), respectively,
- E₁ and E₂ are the energy per photon for \( S_1 \) and \( S_2 \), respectively.
The energy per photon is inversely proportional to the wavelength, so:
E₁ / E₂ = λ₂ / λ₁
where λ₁ = 600 nm and λ₂ = 300 nm.
Therefore, the energy ratio is:
E₁ / E₂ = 300 / 600 = 1 / 2
Substituting this into the power equation:
(N₁ × (1/2)) / N₂ = 2
Simplifying:
N₁ / (2 N₂) = 2 → N₁ = 4 N₂

Step 4: Calculate the Number of Photons for \( S_2 \)

We are given that \( S_1 \) emits \( 2 \times 10^{15} \) photons per second. From the equation \( N_1 = 4 N_2 \), we have:
2 × 10¹⁵ = 4 N₂ → N₂ = (2 × 10¹⁵) / 4 = 0.5 × 10¹⁵
Thus, the number of photons emitted by \( S_2 \) is \( 5 \times 10^{14} \).

Conclusion

The number of photons per second emitted by \( S_2 \) is 5 × 10¹⁴.

Answer 2

Step 1: Understand the relationship between power and the number of photons emitted.
The power of a light source is related to the energy of the photons it emits. The energy of each photon is given by the equation: \[ E = \frac{hc}{\lambda}, \] where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the light.

The power \( P \) of a light source is the rate at which energy is emitted, and it is given by: \[ P = N \times E, \] where \( N \) is the number of photons emitted per second and \( E \) is the energy per photon.
Thus, the power can also be written as: \[ P = N \times \frac{hc}{\lambda}. \] Step 2: Set up the equation for the ratio of powers.
The problem gives us the ratio of the powers of the two light sources \( S_1 \) and \( S_2 \) as: \[ \frac{P_1}{P_2} = 2. \] Also, we know that the power of each source is related to the number of photons emitted and the energy of each photon: \[ \frac{N_1 \times \frac{hc}{\lambda_1}}{N_2 \times \frac{hc}{\lambda_2}} = 2. \] The constants \( hc \) cancel out, and we are left with: \[ \frac{N_1}{N_2} \times \frac{\lambda_2}{\lambda_1} = 2. \] Substituting the known values: \[ \frac{N_1}{N_2} \times \frac{300}{600} = 2. \] Simplifying: \[ \frac{N_1}{N_2} \times \frac{1}{2} = 2 \quad \Rightarrow \quad \frac{N_1}{N_2} = 4. \] Step 3: Find the number of photons emitted by \( S_2 \).
We are given that \( N_1 = 2 \times 10^{15} \) photons per second for \( S_1 \). Using the ratio: \[ \frac{N_1}{N_2} = 4, \] we can find \( N_2 \): \[ N_2 = \frac{N_1}{4} = \frac{2 \times 10^{15}}{4} = 5 \times 10^{14}. \]

Final Answer:
\[ \boxed{5 \times 10^{14}}. \]