91影视

The surface temperature,$T=2000\text{鈥块}$

The intensity per unit wavelength according to classical radiation is $I_c$.

The intensity per unit wavelength according to plank鈥檚 radiation is $I_p$.

The expression for the intensity according to classical radiation is given as follows.

${\mathit{I}}_{\mathbf{c}}\mathbf{=}\frac{\mathbf{2}\mathbf{蟺}\mathbf{c}\mathbf{k}\mathbf{T}}{{\mathbf{位}}^{\mathbf{4}}}$

Here,cis the speed of light, kis the Boltzmann constant and $位$is the wavelength.

The expression for the intensity according to Plank鈥檚 radiation is given as follows.

${\mathit{I}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{2}\mathbf{蟺}{\mathbf{c}}^{\mathbf{2}}\mathbf{h}}{{\mathbf{位}}^{\mathbf{5}}}\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{h}\mathbf{c}\mathbf{/}\mathbf{位}\mathbf{k}\mathbf{T}}\mathbf{鈭�}\mathbf{1}}$

Here,h is the plank鈥檚 constant.

(a)

The value of Boltzmann constant is,$1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}$.

The value of plank鈥檚 constant is,$6.62脳{10}^{-34}\text{鈥�}m\text{鈥�}kg/s$.

Calculate the ratio of $I_c/I_p$.

$\begin{array}{l}\frac{I_c}{I_p}=\frac{\frac{2蟺ckT}{{位}^4}}{\frac{2蟺c^{2}h}{{位}^5}\frac{1}{e^{hc/位kT}鈭�1}}\\ \frac{I_c}{I_p}=\frac{kT}{\frac{hc}{位}\frac{1}{e^{hc/位kT}鈭�1}}\\ \frac{I_c}{I_p}=\frac{位kT}{hc}(e^{hc/位kT}鈭�1)\end{array}$ 鈥�(颈)

Substitute$1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}$for K , $6.62脳{10}^{-34}\text{鈥�}mkg/s$for , for h , $3脳{10}^8$for c ,$2000\text{鈥块}$for T and $400\text{鈥塶尘}$for $位$into equation (i).

$\begin{array}{l}\frac{I_c}{I_p}=\frac{400脳{10}^{鈭�9}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳2000\text{鈥块}}{6.62脳{10}^{-34}\text{鈥�}mkg/s脳3脳{10}^8\text{鈥尘}/\text{s}}\left(e^{\frac{6.62脳{10}^{鈭�34}\text{鈥�}mkg/s脳3脳{10}^8\text{鈥尘}/\text{s}}{400脳{10}^{鈭�9}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳2000\text{鈥块}}}鈭�1\right)\\ \frac{I_c}{I_p}=\frac{1104脳{10}^{鈭�29}}{19.86脳{10}^{鈭�26}}\left(e^{\frac{19.86脳{10}^{鈭�26}}{1104脳{10}^{鈭�29}}}鈭�1\right)\\ \frac{I_c}{I_p}=55.58脳{10}^{鈭�3}(e^{17.989}鈭�1)\\ \frac{I_c}{I_p}=55.58脳{10}^{鈭�3}(6.49脳{10}^7鈭�1)\end{array}$

Solve further as,

$\begin{array}{l}\frac{I_c}{I_p}=0.05558脳6.49脳{10}^7\\ \frac{I_c}{I_p}=0.361脳{10}^7\\ \frac{I_c}{I_p}=3.61脳{10}^6\end{array}$

Hence the ratio of$I_c/I_p$ for wavelength $400\text{鈥塶尘}$is $3.61脳{10}^6$.

(b)

Calculate the ratio of $I_c/I_p$.

Substitute$1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}$for K, $6.62脳{10}^{-34}\text{鈥�}mkg/s$for h , $3脳{10}^8$for c, $2000\text{鈥块}$for T and $200\text{鈥坝纠}$for $位$into equation (i).

$\begin{array}{l}\frac{I_c}{I_p}=\frac{200脳{10}^{鈭�6}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳2000\text{鈥块}}{6.62脳{10}^{-34}\text{鈥�}mkg/s脳3脳{10}^8\text{鈥尘}/\text{s}}\left(e^{\frac{6.62脳{10}^{鈭�34}\text{鈥�}mkg/s脳3脳{10}^8\text{鈥尘}/\text{s}}{200脳{10}^{鈭�6}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳2000\text{鈥块}}}鈭�1\right)\\ \frac{I_c}{I_p}=\frac{552脳{10}^{鈭�26}}{19.86脳{10}^{鈭�26}}\left(e^{\frac{19.86脳{10}^{鈭�26}}{552脳{10}^{鈭�26}}}鈭�1\right)\\ \frac{I_c}{I_p}=27.79(e^{0.0359}鈭�1)\\ \frac{I_c}{I_p}=27.79(1.0366鈭�1)\end{array}$

Solve further as,

$\begin{array}{l}\frac{I_c}{I_p}=27.79脳0.0366\\ \frac{I_c}{I_p}=1.018\end{array}$

Hence the ratio of $I_c/I_p$for wavelength $200\text{鈥�}渭\text{m}$is$1.018$

(c)

From the result of the part (b), it is clear that the classical expression is agree with the Plank鈥檚 expression in the longer wavelength range.

Hence the agreement between $I_c$and $I_p$is at longer wavelength range.