To determine the number of photons striking the plate per square meter per second, we can follow a systematic approach. We need to calculate the energy of each photon and then use the power of the light source to find out how many photons are emitted per second. Finally, we will calculate how many of those photons hit the plate per square meter.
Step 1: Calculate the Energy of a Single Photon
The energy of a photon can be calculated using the formula:
E = h * f
Where:
- E is the energy of the photon.
- h is Planck's constant, approximately 6.626 x 10^-34 J·s.
- f is the frequency of the light.
First, we need to find the frequency (f) using the relationship between wavelength (λ) and frequency:
f = c / λ
Where:
- c is the speed of light, approximately 3 x 10^8 m/s.
- λ is the wavelength, given as 4.8 x 10^-7 m.
Now, substituting the values:
f = (3 x 10^8 m/s) / (4.8 x 10^-7 m) ≈ 6.25 x 10^14 Hz
Next, we can find the energy of a single photon:
E = (6.626 x 10^-34 J·s) * (6.25 x 10^14 Hz) ≈ 4.14 x 10^-19 J
Step 2: Calculate the Total Number of Photons Emitted per Second
The total number of photons emitted per second by the light source can be calculated using the power of the light source:
Number of photons per second = Power / Energy per photon
Given that the power of the light source is 1 watt (which is equivalent to 1 J/s), we can substitute the values:
Number of photons per second = 1 J/s / 4.14 x 10^-19 J ≈ 2.42 x 10^18 photons/s
Step 3: Calculate the Area of the Plate
Assuming the plate has an area of 1 square meter, we can now find the number of photons striking the plate per square meter per second. Since the light falls normally on the plate, the entire power is distributed uniformly over the area.
Step 4: Determine the Number of Photons per Square Meter per Second
Since the plate is 2 meters away from the light source, we need to consider how the intensity of light decreases with distance. The intensity (I) at a distance (r) from a point source is given by:
I = Power / (4πr²)
Substituting the values:
I = 1 W / (4π(2 m)²) ≈ 1 / (16π) W/m² ≈ 0.0199 W/m²
Now, we can find the number of photons striking the plate per square meter per second:
Photons per square meter per second = Intensity / Energy per photon
Photons per square meter per second = 0.0199 W/m² / 4.14 x 10^-19 J ≈ 4.81 x 10^16 photons/m²/s
Final Result
Thus, the number of photons striking the plate per square meter per second is approximately 4.81 x 10^16 photons/m²/s. This calculation illustrates how we can derive the number of photons from fundamental principles of physics, combining concepts of energy, power, and intensity.
