An owl has good night vision because its eyes can detect a light intensity as small as 5.0*10^-13 W/m2. What is the minimum number of photons per second that an owl eye can detect if its pupil has a diameter of 7.0 mm and the light has a wavelength of 545 nm?
To find the minimum number of photons per second that an owl eye can detect, we can use the formula:
Number of photons = (power of light)/(energy per photon)
First, we need to calculate the power of light incident on the owl's eye. We can use the formula for power:
Power = (intensity of light) * (area of the pupil)
Given:
Intensity of light (I) = 5.0 * 10^-13 W/m^2
Diameter of the pupil (d) = 7.0 mm = 7.0 * 10^-3 m
To find the area of the pupil, we can use the formula for the area of a circle:
Area = π * (radius)^2
First, we need to find the radius of the pupil:
Radius (r) = (diameter)/2 = (7.0 * 10^-3)/2 = 3.5 * 10^-3 m
Now, we can calculate the area of the pupil:
Area of the pupil = π * (3.5 * 10^-3)^2 = π * 12.25 * 10^-6 m^2 = 3.1415 * 12.25 * 10^-6 m^2 ≈ 38.484 * 10^-6 m^2
Next, we can calculate the power of light incident on the owl's eye:
Power = (intensity) * (area)
= (5.0 * 10^-13) * (38.484 * 10^-6)
≈ 0.19242 * 10^-18 watts
Now, we need to find the energy per photon. The energy of a photon can be calculated using the formula:
Energy per photon = (Planck's constant * speed of light) / wavelength
Given:
Planck's constant (h) = 6.626 * 10^-34 J·s
Speed of light (c) = 3.0 * 10^8 m/s
Wavelength (λ) = 545 nm = 545 * 10^-9 m
Energy per photon = (6.626 * 10^-34) * (3.0 * 10^8) / (545 * 10^-9)
≈ 3.6441 * 10^-19 J
Finally, we can calculate the minimum number of photons per second:
Number of photons = Power / Energy per photon
= (0.19242 * 10^-18) / (3.6441 * 10^-19)
≈ 0.528
Therefore, the minimum number of photons per second that an owl eye can detect is approximately 0.528 photons/s.
To find the minimum number of photons per second that an owl eye can detect, we need to use the formula:
Number of photons = (Power / Energy of a single photon) x (Area / Time)
1. First, let's calculate the power of the light by using the formula:
Power = Light intensity x Area
Since the light intensity is given as 5.0 x 10^-13 W/m^2, and we want to calculate the power of the light falling on the owl's eye, we need to convert the light intensity to the appropriate units. The area is given by the equation for the area of a circle:
Area = πr^2
where r is the radius of the pupil. In this case, the diameter is given as 7.0 mm, so we convert it to meters by dividing by 1000:
Radius = Diameter / 2 = 7.0 mm / 2 / 1000 = 3.5 mm / 1000 = 3.5 x 10^-3 m
Now, we can calculate the area:
Area = π(3.5 x 10^-3 m)^2 = π(12.25 x 10^-6 m^2)
Plugging the values into the power equation:
Power = (5.0 x 10^-13 W/m^2) x (12.25 x 10^-6 m^2)
2. Next, we need to calculate the energy of a single photon using the formula:
Energy of a single photon = (Planck's constant x speed of light) / wavelength
Planck's constant is denoted by h and has a value of 6.626 x 10^-34 J·s. The speed of light is denoted by c and has a value of 3.0 x 10^8 m/s. The wavelength is given as 545 nm, so we convert it to meters by dividing by 10^9:
Wavelength = 545 nm / 10^9 = 545 x 10^-9 m
Plugging the values into the energy equation:
Energy of a single photon = (6.626 x 10^-34 J·s x 3.0 x 10^8 m/s) / (545 x 10^-9 m)
3. Finally, we can calculate the minimum number of photons per second by using the formula:
Number of photons = (Power / Energy of a single photon) x (Area / Time)
Since we want to find the minimum number of photons per second, we can assume a time interval of 1 second:
Number of photons = (Power / Energy of a single photon) x (Area / 1 s)
Now, we can plug in the calculated values:
Number of photons = (Power) / (Energy of a single photon) x (Area)
Simplifying the equation gives us the minimum number of photons per second that an owl eye can detect.
The energy in one second is what you are seeking.
Energy/second= 5.0E-13W/m^2 * area
Number photons= energy in a second/energy per photon.
The energy per photon= planck's constant*frequency= planck'sconstant*c/wavelength