Question

In an atomic bomb, the temperature of 10 million degrees is developed at the moment of the explosion. In what region of the spectrum does the wavelength corresponding to maximum energy density lie? (\[b = 0.28 \times {10^{ - 2}}\]S.I units)
A) ultraviolet
B) Visible
C) Infra-red rays
D) X-rays

Answer 1

Hint: An atomic bomb gives off radiation of different wavelengths. According to the wavelength, the spectrum lies in different regions. Wien’s displacement law states that the maximum wavelength emitted is inversely proportional to the temperature. This law was given by Wilhelm Wien in 1893. He had given this law for the blackbody radiation.

Formula used:
\[{\lambda _m} = \dfrac{b}{T}\]
Where,
\[{\lambda _m}\]=maximum wavelength
\[b\]=proportionality constant
\[T\]=absolute temperature

Complete step by step answer:
In an atomic bomb, the temperature of 10 million degrees is emitted at the moment of explosion. According to the temperature emitted, a particular wavelength for the spectrum can be found by
\[{\lambda _m} = \dfrac{b}{T}\]
As the temperature is 10 million degrees, temperature \[T = 10 \times {10^6}\] degrees. Which is further changed when transformed in to kelvin as \[T = {10^7} + 273K\] which is approximately equal to \[T = {10^7}K\]
Now applying the values in the known formula for maximum wavelength. Therefore,
\[ \Rightarrow {\lambda _m} = \dfrac{{0.28 \times {{10}^{ - 2}}}}{{{{10}^7}}}\]
We can cancel out the power terms as the base is same number, we get
\[ \Rightarrow {\lambda _m} = 0.28 \times {10^{ - 9}}m\]
\[\therefore {\lambda _m} = 0.28nm\]
From the value found, we can say that the wavelength of the spectrum is nanometre.
Therefore, the spectrum lies in the ultraviolet region. As the wavelength of the ultraviolet rays is in nanometre.

Therefore, the correct option is A.

Note:
The wavelength and the frequency are inversely proportional to each other. If there is a maximum wavelength, the frequency decreases, and if there is a minimum wavelength, the frequency increases. The radiation can be represented by energy, frequency, and wavelength. Each electromagnetic radiation has its own wavelength, frequency. Ultraviolet rays have maximum frequency among the electro-magnetic radiations.