If the power of light is 10 kW and the wavelength is 300 nm, then find the number of photons per second.

Steps to Solve

1. Understand the energy of a single photon

The energy of a single photon can be calculated 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 ($3.00 \times 10^8 \text{ m/s}$)
• $\lambda$ is the wavelength in meters

First, convert the wavelength from nanometers to meters: $$ \lambda = 300 \text{ nm} = 300 \times 10^{-9} \text{ m} $$

2. Calculate the energy of a single photon

Substituting the values into the energy formula: $$ E = \frac{(6.626 \times 10^{-34} \text{ Js})(3.00 \times 10^{8} \text{ m/s})}{300 \times 10^{-9} \text{ m}} $$

3. Calculate total energy emitted per second

The power of the light source is given as 10 kW, which is equal to: $$ P = 10 \text{ kW} = 10,000 \text{ W} $$ Since power is energy per unit time, the total energy emitted per second is simply the power: $$ \text{Total Energy} = P = 10,000 \text{ J/s} $$

4. Calculate the number of photons emitted per second

Finally, the number of photons emitted per second can be calculated using: $$ N = \frac{\text{Total Energy}}{E} $$ Where $N$ is the number of photons.

Substituting the values: $$ N = \frac{10,000 \text{ J/s}}{E} $$

5. Complete the calculation for N

Now plug in the value for $E$ from step 2 to find $N$.