To solve the problem of how many photons of green light are needed to generate a minimum energy of \(10^{-17}\) J, we will follow these steps:
### Step 1: Understand the relationship between energy, frequency, and wavelength
The energy of a single photon can be calculated using the formula:
\[
E = h \nu
\]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.6 \times 10^{-34} \, \text{Js}\)),
- \(\nu\) is the frequency of the light.
### Step 2: Relate frequency to wavelength
The frequency \(\nu\) can be expressed in terms of the speed of light \(c\) and wavelength \(\lambda\) using the equation:
\[
c = \nu \lambda
\]
Rearranging this gives:
\[
\nu = \frac{c}{\lambda}
\]
### Step 3: Substitute frequency into the energy equation
Substituting \(\nu\) into the energy equation gives:
\[
E = h \left(\frac{c}{\lambda}\right)
\]
This can be rearranged to find the energy of a photon in terms of wavelength:
\[
E = \frac{hc}{\lambda}
\]
### Step 4: Calculate the energy of a single photon of green light
Given:
- Wavelength \(\lambda = 495 \, \text{nm} = 495 \times 10^{-9} \, \text{m}\)
- Speed of light \(c = 3 \times 10^8 \, \text{m/s}\)
Now, substituting the values into the energy equation:
\[
E = \frac{(6.6 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{495 \times 10^{-9} \, \text{m}}
\]
### Step 5: Calculate the energy
Calculating the numerator:
\[
6.6 \times 10^{-34} \times 3 \times 10^8 = 1.98 \times 10^{-25} \, \text{Jm}
\]
Now, calculating the energy:
\[
E = \frac{1.98 \times 10^{-25}}{495 \times 10^{-9}} = \frac{1.98 \times 10^{-25}}{4.95 \times 10^{-7}} = 4.00 \times 10^{-19} \, \text{J}
\]
### Step 6: Calculate the number of photons needed
To find the number of photons \(n\) required to generate \(10^{-17} \, \text{J}\):
\[
n = \frac{E_{\text{total}}}{E_{\text{photon}}} = \frac{10^{-17} \, \text{J}}{4.00 \times 10^{-19} \, \text{J}}
\]
Calculating \(n\):
\[
n = \frac{10^{-17}}{4.00 \times 10^{-19}} = 25
\]
### Final Answer
Thus, the number of photons of green light needed is:
\[
\boxed{25}
\]
