A sphere with emissivity 0.613 and radius of 0.841 m has a temperature of 58.7°C. If the sphere is placed inside an environment at temperature 89.2°C, what is the net rate of energy exchange? Calculate the power of absorption and subtract the power of radiation to give the net power. Use Watts (W) as your unit and give the unit.

To solve this problem, we need to use the Stefan-Boltzmann law, which states that the power radiated by a black body in watts is proportional to the fourth power of the temperature of that body in Kelvin. The formula is:

P = εσAT^4

where:

• P is the power in watts,
• ε is the emissivity of the body (dimensionless),
• σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2K^4),
• A is the surface area of the body in square meters, and
• T is the temperature of the body in Kelvin.

First, we need to convert the temperatures from Celsius to Kelvin. The conversion formula is T(K) = T(°C) + 273.15.

T1 = 58.7°C + 273.15 = 331.85 K (temperature of the sphere) T2 = 89.2°C + 273.15 = 362.35 K (temperature of the environment)

Next, we calculate the surface area of the sphere. The formula for the surface area of a sphere is A = 4πr^2.

A = 4π(0.841 m)^2 = 8.89 m^2

Now we can calculate the power of radiation and absorption.

The power of radiation (Pr) is the power radiated by the sphere:

Pr = εσAT1^4 = 0.613 * 5.67 x 10^-8 W/m^2K^4 * 8.89 m^2 * (331.85 K)^4 = 1053.6 W

The power of absorption (Pa) is the power absorbed by the sphere from the environment:

Pa = εσAT2^4 = 0.613 * 5.67 x 10^-8 W/m^2K^4 * 8.89 m^2 * (362.35 K)^4 = 1321.6 W

Finally, the net rate of energy exchange is the power of absorption minus the power of radiation:

Pnet = Pa - Pr = 1321.6 W - 1053.6 W = 268 W

So, the net rate of energy exchange is 268 Watts.