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?

Options

A. 14
B. 28
C. 39
D. 42

Answer

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 Energy from Joules to Electron Volts
The energy required is given as \(10^{-17} J\). To convert this energy into electron volts (eV), we use the conversion factor \(1 eV = 1.6 \times 10^{-19} J\).

\[ E (in \, eV) = \frac{10^{-17} J}{1.6 \times 10^{-19} J/eV} = 0.625 \times 10^{2} eV = 62.5 eV \]

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

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

Where:
- \(E\) is the energy of one photon,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} J \cdot s\)),
- \(c\) is the speed of light (\(3 \times 10^{8} m/s\)),
- \(\lambda\) is the wavelength of light (in meters).

### Step 3: Convert Wavelength from Nanometers to Meters
The wavelength of green light is given as \(550 nm\). To convert this to meters:

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

### Step 4: Calculate the Energy of One Photon
Now, we can calculate the energy of one photon of green light:

\[ E = \frac{(6.626 \times 10^{-34} J \cdot s)(3 \times 10^{8} m/s)}{550 \times 10^{-9} m} \]

Calculating this gives:

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

### Step 5: Calculate the Number of Photons Required
To find the number of photons \(n\) needed to achieve the total energy of \(10^{-17} J\):

\[ n = \frac{Total \, Energy}{Energy \, per \, Photon} = \frac{10^{-17} J}{3.61 \times 10^{-19} J} \]

Calculating this gives:

\[ n \approx 27.7 \]

Since the number of photons must be a whole number, we round this to:

\[ n \approx 28 \]

### Conclusion
Therefore, the minimum number of photons of green light needed to generate the required energy is **28**.

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