Question

A$2.0-cm-$diameter metal sphere is glowing red, but a spectrum shows that its emission spectrum peaks at an infrared wavelength of $2.0mm$.How much power does the sphere radiate? Assume$e=1.$

Short answer

The power that sphere radiates is $3.1\times {10}^{-10}Watt$

Step-by-step solution

Given Information

We are given that diameter of metal sphere is$2.0-cm$ , infrared wavelength of spectrum is $2.0mm$ and $e=1$.

We are asked to find power that the sphere radiates.

Simplify

We are given with the diameter of metal sphere , we will find radius of sphere

$r=\frac{d}{2}=\frac{2cm}{2}=1cm$

We will convert cm into m $\Rightarrow 1cm=0.01m$

According to Wien 's law $\Rightarrow \lambda =\frac{2.9\times {10}^6(nm.k)}{T}$

We can find the value of T from this equation, $T=\frac{2.90\times {10}^6(nm.k)}{2\times {10}^{6}nm}=1.45K$

now power radiated by sphere is given by ,

Power $=e\sigma A{T}^4$$=1\times \left(5.67\times {10}^{-8}\right)\times 4\mathrm{\pi }\times {\left(0.01\right)}^2\times {\left(1.45\right)}^4$$=3.1\times {10}^{-10}Watt$