A P-N junction becomes active as photons of wavelength, `lambda=400 nm` falls on it. Find the energy band gap?

Options

A. `3.09 eV`
B. `4.51eV`
C. `2.45eV`
D. `5.34eV`

Answer

To find the energy band gap of a P-N junction when photons of wavelength \( \lambda = 400 \, \text{nm} \) fall on it, we can use the formula that relates energy to wavelength:

### Step-by-Step Solution:

**Step 1: Understand the relationship between energy and wavelength.**
The energy \( E \) of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the photon.

**Step 2: Convert the wavelength from nanometers to meters.**
Given \( \lambda = 400 \, \text{nm} \):
\[ \lambda = 400 \times 10^{-9} \, \text{m} \]

**Step 3: Substitute the values into the energy formula.**
Now, substituting the values of \( h \), \( c \), and \( \lambda \) into the energy formula:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{Js}) \times (3 \times 10^8 \, \text{m/s})}{400 \times 10^{-9} \, \text{m}} \]

**Step 4: Calculate the energy in joules.**
Calculating the above expression:
\[ E = \frac{(6.626 \times 3) \times 10^{-26}}{400 \times 10^{-9}} = \frac{19.878 \times 10^{-26}}{400 \times 10^{-9}} = \frac{19.878 \times 10^{-26}}{4 \times 10^{-7}} = 4.9695 \times 10^{-19} \, \text{J} \]

**Step 5: Convert the energy from joules to electronvolts (eV).**
To convert joules to electronvolts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E \, (\text{in eV}) = \frac{4.9695 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 3.1059 \, \text{eV} \]

**Step 6: Round the answer.**
Rounding to two decimal places, we find:
\[ E \approx 3.09 \, \text{eV} \]

### Final Answer:
The energy band gap is approximately \( 3.09 \, \text{eV} \).

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