1. Use power = energy emitted per second

For monochromatic light, energy of one photon is $E = \frac{hc}{\lambda}$

If the source emits $N$ photons per second, then power is $P = N E = N\frac{hc}{\lambda}$

Since the source operates at the same power, we have $N \propto \lambda$

because $N = \frac{P\lambda}{hc}$

2. Relate the two cases

Given:

• At $\lambda_1 = 600\,\text{nm}$, $N_1 = 9 \times 10^{20}\,\text{photons/s}$
• At $\lambda_2 = 800\,\text{nm}$, find $N_2$

Using proportionality: $\frac{N_2}{N_1} = \frac{\lambda_2}{\lambda_1}$

So, $N_2 = N_1 \cdot \frac{\lambda_2}{\lambda_1}$

Substitute values: $N_2 = 9 \times 10^{20} \cdot \frac{800}{600}$

$N_2 = 9 \times 10^{20} \cdot \frac{4}{3}$

$N_2 = 12 \times 10^{20}$

3. Match with options

The correct option is: $\boxed{\text{A: } 12 \times 10^{20}}$

4. Comparison with stored answer

Stored correct answer: A

Our derived answer is also A. So they agree.