Q: A sphere with diameter of 80 cm is held at a temperature of 250°C and is radiating energy. If the intensity of the radiation detected at a distance of 2.0 m from the sphere’s centre is 102 W/m², What is the emissivity of the sphere?

Sol: From inverse square law $\large I \propto \frac{1}{r^2}$

$\large \frac{I_2}{I_1} = \frac{r_1^2}{r_2^2}$

$\large I_2 = I_1 (\frac{r_2^2}{r_1^2}) = 102 (\frac{2}{0.4})^2 $

= 2.55 × 10^{3} W/m^{2}

The power at the surface of the sphere

P = I_{2} (4π r_{2}^{2} ) = 2.55 × 10^{3} × 4π(0.4)^{2}

and , P = e σ A T^{4}

2.55 × 10^{3} × 4π(0.4)^{2} = e × 5.67 × 10^{-8} × 4 π(0.4)^{2} ×(523)^{4}

e = 0.61