91影视

The surface temperature is,$T=98.6掳\text{鈥塅}$

The surface area is,$A=4{\text{鈥塩尘}}^2$.

The wavelength range is, $螖位=1\text{鈥塶尘}$.

The expression of Wein鈥檚 displacement law to calculate the wavelength is given as follows.

${位}_{max}T=2898\text{鈥坝纠.K}$ 鈥�(颈)

The expression to calculate the spectral radiancy is given as follows.

$\mathit{S}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}\mathbf{蟺}{\mathbf{c}}^{\mathbf{2}}\mathbf{h}}{{\mathbf{位}}^{\mathbf{5}}}\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{h}\mathbf{c}\mathbf{/}\mathbf{位}\mathbf{k}\mathbf{T}}\mathbf{鈭�}\mathbf{1}}$ 鈥�(颈颈)

Here, his the plank鈥檚 constant, cis the speed of light, k is the Boltzmann constant and$位$is the wavelength.

The expression to calculate the radiated power is given as follows.

$\mathit{P}\mathbf{=}\mathit{S}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathit{A}\mathit{螖}\mathit{位}$ 鈥�(颈颈颈)

Here,$\mathit{S}\mathbf{\left(}\mathbf{位}\mathbf{\right)}$is the spectral radiancy.

The expression to calculate the rate of emitted photons per second is given as follows.

$\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}\mathbf{=}\frac{\mathbf{位}\mathbf{P}}{\mathbf{h}\mathbf{c}}$ 鈥�(颈惫)

(a)

Calculate the temperature from Fahrenheit to kelvin,

$\begin{array}{l}T=(98.6\text{鈥塅}鈭�32)\text{鈥塁}脳\frac{5}{9}+273.15\text{鈥块}\\ T=37\text{鈥塁}+273.15\text{鈥块}\\ T=310.15\text{鈥块}\end{array}$

Calculate the wavelength,

Substitute$310.15\text{鈥块}$ for T into equation (i).

$\begin{array}{l}{位}_{\max }脳310.15\text{鈥块}=2898\text{鈥坝纠.K}\\ {位}_{\max }=\frac{2898\text{鈥坝纠.K}}{310.15\text{鈥块}}\\ {位}_{\max }=9.343\text{鈥坝纠}\end{array}$

Hence the maximum value of wavelength at which spectral radiancy is maximum is $9.343\text{鈥坝纠}$.

(b)

Calculate the spectral radiancy.

Substitute $1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}$for K ,$6.62脳{10}^{-34}\text{鈥�}m\text{鈥�}kg/s$ for h,$3脳{10}^8\text{鈥尘}/{\text{c}}^{\text{2}}$ for c, $310.15\text{鈥块}$for T and$9.343\text{鈥坝纠}$ for$位$ into equation (i).

$\begin{array}{l}s\left(位\right)=\frac{2蟺{(3脳{10}^8\text{鈥尘}/{\text{c}}^{\text{2}})}^2脳6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}}{{(9.343脳{10}^{鈭�6}\text{鈥尘})}^5}脳\frac{1}{e^{\frac{6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}脳3脳{10}^8\text{鈥尘}/{\text{c}}^{\text{2}}}{9.343脳{10}^{鈭�6}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳310.15\text{鈥块}}}鈭�1}\\ s\left(位\right)=\frac{374.3521脳{10}^{鈭�18}{\text{鈥尘}}^{\text{3}}\text{.kg}/{\text{s.c}}^{\text{4}}}{7.119脳{10}^{鈭�26}{\text{鈥尘}}^{\text{5}}}脳7.00脳{10}^{鈭�3}\\ s\left(位\right)=3.67脳{10}^7\text{鈥塛}/{\text{m}}^3\end{array}$

Calculate the radiated power.

Substitute$3.67脳{10}^7\text{鈥塛}/{\text{m}}^3$ for$S\left(位\right)$,$1\text{鈥塶尘}$ for $螖位$and$4{\text{鈥塩尘}}^2$ for A into equation (iii).

$\begin{array}{l}P=3.67脳{10}^7\text{鈥塛}/{\text{m}}^{\text{3}}脳4脳{10}^{鈭�4}{\text{鈥尘}}^{\text{2}}脳1脳{10}^{鈭�9}\text{鈥尘}\\ P=14.68脳{10}^{鈭�6}\text{鈥塛}\\ P=1.468脳{10}^{鈭�5}\text{鈥塛}\end{array}$

Hence the radiated power is $1.468脳{10}^{鈭�5}\text{鈥塛}$.

(c)

Calculate the rate of emitted photons.

Substitute $1.468脳{10}^{鈭�5}\text{鈥塛}$for P ,$9.343\text{鈥坝纠}$ for $位$, $6.62脳{10}^{-34}\text{鈥�}m\text{鈥�}kg/s$for h, and$3脳{10}^8\text{鈥尘}/\text{s}$ for into equation (iv).

$\begin{array}{l}\frac{dN}{dt}=\frac{9.343脳{10}^{鈭�6}\text{鈥尘}脳1.468脳{10}^{鈭�5}\text{鈥塛}}{6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}脳3脳{10}^8\text{鈥尘}/\text{s}}\\ \frac{dN}{dt}=6.9脳{10}^{14}\text{鈥塸丑辞迟辞苍蝉}/\text{s}\end{array}$

Hence the rate of emitted photons per second is $6.9脳{10}^{14}\text{鈥塸丑辞迟辞苍蝉}/\text{s}$.

(d)

The value of the wavelength,$位=500\text{鈥塶尘}$

Calculate the spectral radiancy.

Substitute $1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}$for k , $6.62脳{10}^{-34}\text{鈥�}m\text{鈥�}kg/s$for h , $3脳{10}^8\text{鈥尘}/\text{s}$for c,$310.15\text{鈥块}$ for T and $500\text{鈥塶尘}$for$位$ into equation (i).

$\begin{array}{l}s\left(位\right)=\frac{2蟺{(3脳{10}^8\text{鈥尘}/{\text{c}}^{\text{2}})}^2脳6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}}{{(500脳{10}^{鈭�9}\text{鈥尘})}^5}脳\frac{1}{e^{\frac{6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}脳3脳{10}^8\text{鈥尘}/{\text{c}}^{\text{2}}}{500脳{10}^{鈭�9}\text{鈥尘}脳1.38脳{10}^{鈭�23}\text{鈥塉}/\text{K}脳310.15\text{鈥块}}}鈭�1}\\ s\left(位\right)=\frac{374.3521脳{10}^{鈭�18}{\text{鈥尘}}^{\text{3}}\text{.kg}/{\text{s.c}}^{\text{4}}}{3.125脳{10}^{鈭�32}{\text{鈥尘}}^{\text{5}}}脳4.65脳{10}^{鈭�41}\\ s\left(位\right)=5.56脳{10}^{鈭�25}\text{鈥塛}/{\text{m}}^3\end{array}$

Calculate the radiated power.

Substitute$5.56脳{10}^{鈭�25}\text{鈥塛}/{\text{m}}^3$ for $S\left(位\right)$, $1\text{鈥塶尘}$for $螖位$and$4{\text{鈥塩尘}}^2$ for A into equation (iii).

$\begin{array}{l}P=5.56脳{10}^{鈭�25}\text{鈥塛}/{\text{m}}^{\text{3}}脳4脳{10}^{鈭�4}{\text{鈥尘}}^{\text{2}}脳1脳{10}^{鈭�9}\text{鈥尘}\\ P=2.22脳{10}^{鈭�37}\text{鈥塛}\end{array}$

Hence the radiated power is$2.22脳{10}^{鈭�37}\text{鈥塛}$

(e)

Calculate the rate of emitted photons.

Substitute$2.22脳{10}^{鈭�37}\text{鈥塛}$ for P, $500\text{鈥塶尘}$for $位$, $6.62脳{10}^{-34}\text{鈥�}m\text{鈥�}kg/s$for h,and $3脳{10}^8\text{鈥尘}/\text{s}$for c into equation (iv).

$\begin{array}{l}\frac{dN}{dt}=\frac{500脳{10}^{鈭�9}\text{鈥尘}脳2.22脳{10}^{鈭�37}\text{鈥塛}}{6.62脳{10}^{鈭�34}\text{鈥尘kg}/\text{s}脳3脳{10}^8\text{鈥尘}/\text{s}}\\ \frac{dN}{dt}=5.58脳{10}^{鈭�19}\text{鈥塸丑辞迟辞苍蝉}/\text{s}\end{array}$

Hence the rate of emitted photons per second is$5.58脳{10}^{鈭�19}\text{鈥塸丑辞迟辞苍蝉}/\text{s}$.