Question

A heated body emits radiation which has maximum ${v}_{0}$. The emissivity of the material is 0.5. If the absolute temperature of the body is doubled. Then:
A. the maximum intensity of radiation will be near the frequency $2{v}_{0}$.
B. the maximum intensity of radiation will be near the frequency $\dfrac {{v}_{0}}{2}$.
C. the total energy emitted will increase by a factor of 32.
D. the total energy emitted will increase by a factor of 8.

Answer 1

Hint: To find the solution for this question, use the equation for Wein displacement law for maximum intensity. Then, use the relationship between frequency and wavelength. Now, compare both the equations mentioned above. After comparing the equations, find the relationship between absolute temperature and frequency of radiation. Now, check what happens to frequency when the absolute temperature of the body is doubled.

Formula used:
${\lambda}_{0} T= constant$
${\lambda}_{0} \propto \dfrac {1}{{v}_{0}}$

Complete answer:
According to Wein displacement law, at maximum intensity,
${\lambda}_{0} T= constant$ …(1)
Where, ${\lambda}_{0}$ is the maximum wavelength
             T is the absolute temperature
From the equation. (1) we can write,
${\lambda}_{0} \propto \dfrac {1}{T}$ …(2)
We know, the relationship between frequency and wavelength is given by,
${\lambda}_{0} \propto \dfrac {1}{{v}_{0}}$ …(3)
Where, ${v}_{0}$ is the maximum frequency
Comparing equation. (2) and (3) we get,
$\dfrac {1}{T} \propto \dfrac {1}{{v}_{0}}$
$\Rightarrow T \propto {v}_{0}$ …(4)
From the equation. (4) we can infer that the absolute temperature is directly proportional to the frequency.
If the absolute temperature is doubled then the frequency also gets doubled.
Thus, the maximum intensity of radiation will be near the frequency $2{v}_{0}$.

So, the correct answer is option A i.e. the maximum intensity of radiation will be near the frequency $2{v}_{0}$.

Note:
From the above solution we can observe that the frequency and wavelength is independent of the value of emissivity. But the total energy radiated per unit area does depend on the emissivity of the material. If we have to calculate the energy emitted per unit area then we can use the Stefan-Boltzmann law which is given by,
$E=\sigma \varepsilon { T }^{ 4 }$
Where, E is the energy emitted
             $\sigma$ is the Stefan-Boltzmann constant
             $\varepsilon$ is the emissivity of the material
             T is the absolute temperature of the material