Imagine a space heater consisting of an iron ball (radius 10.0 cm) through which electrical current is passed in order to heat the ball. What temperature would it need to be in order to radiate a net power of 500 W into the surrounding air (which is at a temperature of 20°C)? Assume the ball is a perfect blackbody.

Question

Quizplus · Accepted Answer

The net power radiated by the ball can be calculated using the Stefan-Boltzmann law for a blackbody, which is $P = \sigma A (T^4 - T_{\text{env}}^4)$, where $P$ is the power, $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}$), $A$ is the surface area of the ball, $T$ is the temperature of the ball in Kelvin, and $T_{\text{env}}$ is the temperature of the environment in Kelvin. The surface area $A$ of a sphere is $4\pi r^2$, where $r$ is the radius of the sphere.Given:- $r = 10.0 \, \text{cm} = 0.1 \, \text{m}$- $P = 500 \, \text{W}$- $T_{\text{env}} = 20^\circ\text{C} = 293 \, \text{K}$First, calculate the surface area $A$:$$A = 4\pi r^2 = 4\pi (0.1)^2 = 4\pi (0.01) \, \text{m}^2$$Then, rearrange the Stefan-Boltzmann law to solve for $T$:$$T = \left(\frac{P}{\sigma A} + T_{\text{env}}^4\right)^{1/4}$$Substitute the given values and calculate $T$:$$T = \left(\frac{500}{5.67 \times 10^{-8} \cdot 4\pi (0.01)} + 293^4\right)^{1/4}$$$$T \approx 530 \, \text{K}$$Therefore, the temperature of the ball needs to be approximately 530 K to radiate a net power of 500 W into the surrounding air.

Imagine a Space Heater Consisting of an Iron Ball (Radius