Question
Use the formula $\lambda _m T = 0.29 \ cm K$ to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?
✓
Answer
A body at a particular temperature produces a continous spectrum of wavelengths. In case of a black body, the wavelength corresponding to maximum intensity of radiation is given according to Planck's law. It can be given by the relation,
$\lambda_\text{m}=\frac{0.29}{\text{T}}\text{ cm} \ \text{K}$
Where,
$\lambda_\text{m}=$ maximum wavelength
$T =$ temperature
Thus, the temperature for different wavelengths can be obtained as:
$\text{For} \ \lambda_\text{m}=10^{-4} \ \text{cm}; \ \text{T}=\frac{0.29}{10^{-4}}=2900\ ^\circ\text{K}$
$\text{For} \ \lambda_\text{m}=5\times10^{-5} \ \text{cm}; \ \text{T}=\frac{0.29}{5\times10^{-5}}=5800\ ^\circ\text{K}$
$\text{For} \ \lambda_\text{m}=10^{-6} \ \text{cm}; \ \text{T}=\frac{0.29}{10^{-6}}=290000\ ^\circ\text{K}$ and so on.
The numbers obtained tell us that temperature ranges are required for obtaining radiations in different parts of an electromagnetic spectrum.
As the wavelength decreases, the corresponding temperature Increases.
$\lambda_\text{m}=\frac{0.29}{\text{T}}\text{ cm} \ \text{K}$
Where,
$\lambda_\text{m}=$ maximum wavelength
$T =$ temperature
Thus, the temperature for different wavelengths can be obtained as:
$\text{For} \ \lambda_\text{m}=10^{-4} \ \text{cm}; \ \text{T}=\frac{0.29}{10^{-4}}=2900\ ^\circ\text{K}$
$\text{For} \ \lambda_\text{m}=5\times10^{-5} \ \text{cm}; \ \text{T}=\frac{0.29}{5\times10^{-5}}=5800\ ^\circ\text{K}$
$\text{For} \ \lambda_\text{m}=10^{-6} \ \text{cm}; \ \text{T}=\frac{0.29}{10^{-6}}=290000\ ^\circ\text{K}$ and so on.
The numbers obtained tell us that temperature ranges are required for obtaining radiations in different parts of an electromagnetic spectrum.
As the wavelength decreases, the corresponding temperature Increases.
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