A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is 1.42 eV. The wavelength of light emitted from the LED is:
- 650 nm
- 1243 nm
- 875 nm
- 1400 nm
The Correct Option is C
Approach Solution - 1
To determine the wavelength of light emitted from the LED, we need to understand the relationship between the energy of a photon and its wavelength. The energy of a photon is given by the equation:
\(E = \frac{hc}{\lambda}\)
Where:
- \(E\) is the energy of the photon, measured in electron volts (eV).
- \(h\) is Planck's constant, approximately \(4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s}\).
- \(c\) is the speed of light in vacuum, approximately \(3 \times 10^{8} \, \text{m/s}\).
- \(\lambda\) is the wavelength of the emitted light, measured in meters.
The band gap energy of GaAs (Gallium Arsenide) is given as \(1.42 \, \text{eV}\). This energy corresponds to the energy of the photons emitted from the LED.
We will rearrange the equation to solve for wavelength \(\lambda\):
\(\lambda = \frac{hc}{E}\)
Substitute the known values into the equation:
\(\lambda = \frac{4.1357 \times 10^{-15} \, \text{eV} \cdot \text{s} \times 3 \times 10^{8} \, \text{m/s}}{1.42 \, \text{eV}}\)
Calculate the wavelength:
\(\lambda = \frac{12.4071 \times 10^{-7} \, \text{m} \, \text{eV/s}}{1.42 \, \text{eV}} \approx 8.7408 \times 10^{-7} \, \text{m}\)
Convert meters to nanometers (1 m = 109 nm):
\(\lambda \approx 874.08 \, \text{nm}\)
The wavelength of the light emitted from the LED is approximately \(875 \, \text{nm}\).
This calculation matches the correct answer, which is \(875 \, \text{nm}\).
Therefore, the correct answer is 875 nm.
Approach Solution -2
The wavelength of light emitted from an LED is related to its energy gap by the formula:
\( \lambda = \frac{1240}{E_g} \),
where:
- \( \lambda \) is the wavelength in nanometers (nm),
- \( E_g \) is the energy gap in electron volts (eV).
Substitute \( E_g = 1.42 \, \text{eV} \):
\( \lambda = \frac{1240}{1.42} \).
Perform the calculation: \( \lambda = 875 \, \text{nm} \, \text{(approximately)}. \)
Final Answer: 875 nm.
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