Question

A black body at $1227^\circ {\rm{C}}$ emits radiations with maximum intensity at a wavelength of $5000\;\mathop {\rm{A}}\limits^ \circ$ . The temperature of the body is increased by $1000^\circ {\rm{C}}\;$, the maximum intensity will be observe at:
(A) $4000\;\mathop {\rm{A}}\limits^ \circ$
(B) $5000\;\mathop {\rm{A}}\limits^ \circ$
(C) $6000\;\mathop {\rm{A}}\limits^ \circ$
(D) $3000\;\mathop {\rm{A}}\limits^ \circ$

Answer 1

In the question, we have given the black body's temperature to emit the radiation of the maximum intensity of a particular wavelength. We need to find the wavelength for which the black body will emit radiation of maximum intensity at increased temperature. We will use Wein's displacement law to find the relation between wavelength and temperature of the radiation.

Complete step by step answer:
Given :
The temperature of the black body is ${T_1} = 1227^\circ {\rm{C}}$ .
The increased temperature of the body is $1000^\circ {\rm{C}}\;$.
At temperature ${T_1}$ , the black body emits radiation of wavelength ${\lambda _1} = 5000\;\mathop {\rm{A}}\limits^ \circ$.
When the temperature is increased by $1000^\circ {\rm{C}}\;$, we will write it as:

{T_2} = 1227^\circ {\rm{C}} + 1000^\circ {\rm{C}}\\\ {T_2} = 2227^\circ {\rm{C}} \end{array}$$ We will convert ${T_1}$ and ${T_2}$ into kelvin. This can be expressed as: $\begin{array}{l} {T_1} = \left( {1227 + 273} \right)\;{\rm{K}}\\\ {T_1} = 1500\;{\rm{K}} \end{array}$ And $\begin{array}{l} {T_2} = \left( {2227 + 273} \right)\;{\rm{K}}\\\ {T_2} = 2500\;{\rm{K}} \end{array}$ From Wein's displacement law concept, the curve of spectral energy density and temperature of a black body will give different peak points. It is inversely proportional to the absolute temperature of the black body. Since, from the Wein's displacement law, we also know that $\lambda \cdot T = 2.898 \times {10^{ - 3}}\;{\rm{m}} \cdot {\rm{K}}$. Hence, we can say that, ${\lambda _1} \cdot {T_1} = {\lambda _2} \cdot {T_2}$ On substituting $$5000\;\mathop {\rm{A}}\limits^ \circ $$ for $${\lambda _1}$$, $1500\;{\rm{K}}$ for ${T_1}$ , and $2500\;{\rm{K}}$ for ${T_2}$ in the above expression to find out the wavelength at the temperature ${T_2}$. $\begin{array}{l} \left( {5000\;\mathop {\rm{A}}\limits^ \circ } \right) \cdot \left( {1500\;{\rm{K}}} \right) = {\lambda _2} \cdot \left( {2500\;{\rm{K}}} \right)\\\ {\lambda _2} = 3000\;\mathop {\rm{A}}\limits^ \circ \end{array}$ Therefore, the maximum intensity for the increased temperature will be observed at $3000\;\mathop {\rm{A}}\limits^ \circ $. Hence option D is correct. **Note:** In the question, the temperature given in the question is in Celsius, but we need to substitute temperature in Kelvin for the Wein's displacement law. Make sure to convert temperature given Celsius into kelvin.