Suppose `10^(-17)J` of light energy is needed by the interior of the human eye to see an object. How many photons of green light `(lambda=550nm)` are needed to generate this minimum amount of energy?

Explanation

To solve the problem of how many photons of green light (with a wavelength of 550 nm) are needed to generate an energy of \(10^{-17} J\), we can follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength of green light is \(550 nm\). We need to convert this to meters for our calculations.

\[ \lambda = 550 \, nm = 550 \times 10^{-9} \, m \]

### Step 2: Use the Energy of a Photon Formula
The energy \(E\) of a single photon can be calculated using the formula:

\[ E = \frac{hc}{\lambda} \]

Where:
- \(h\) is Planck's constant, \(h = 6.626 \times 10^{-34} \, J \cdot s\)
- \(c\) is the speed of light, \(c = 3.00 \times 10^8 \, m/s\)
- \(\lambda\) is the wavelength in meters.

### Step 3: Calculate the Energy of One Photon
Substituting the values into the formula:

\[ E = \frac{(6.626 \times 10^{-34} \, J \cdot s)(3.00 \times 10^8 \, m/s)}{550 \times 10^{-9} \, m} \]

Calculating this gives:

\[ E \approx \frac{1.9878 \times 10^{-25}}{550 \times 10^{-9}} \approx 3.61 \times 10^{-19} \, J \]

### Step 4: Calculate the Number of Photons Needed
Now, we need to find out how many photons are required to generate \(10^{-17} J\) of energy. We can use the formula:

\[ N = \frac{E_{total}}{E_{photon}} \]

Where:
- \(E_{total} = 10^{-17} J\)
- \(E_{photon} = 3.61 \times 10^{-19} J\)

Substituting the values:

\[ N = \frac{10^{-17}}{3.61 \times 10^{-19}} \approx 277.85 \]

### Step 5: Round to the Nearest Whole Number
Since we cannot have a fraction of a photon, we round this to the nearest whole number:

\[ N \approx 278 \, \text{photons} \]

### Final Answer
The number of photons of green light needed to generate \(10^{-17} J\) of energy is approximately **278 photons**.

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