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 equal to the Stefan-Boltzmann constant times the surface area of the object times the temperature to the fourth power. The equation is:

P = σ * A * T^4

where:
P is the power radiated,
σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W/m^2K^4),
A is the surface area of the object, and
T is the temperature in Kelvin.

We are given that P = 500 W and we need to find T.

First, we need to calculate the surface area of the iron ball. The surface area of a sphere is given by the equation A = 4πr^2. The radius r is given as 10 cm, but we need to convert this to meters, so r = 0.1 m.

A = 4π(0.1 m)^2 = 0.04π m^2

Now we can substitute P, σ, and A into the Stefan-Boltzmann Law and solve for T:

500 W = (5.67 * 10^-8 W/m^2K^4) * (0.04π m^2) * T^4

Solving for T gives:

T^4 = 500 W / [(5.67 * 10^-8 W/m^2K^4) * (0.04π m^2)]

T^4 = 2.21 * 10^9 K^4

Taking the fourth root of both sides gives:

T = (2.21 * 10^9 K^4)^(1/4) = 728 K

However, this is the temperature in Kelvin. To convert to Celsius, we subtract 273.15:

T = 728 K - 273.15 = 454.85 °C

So, the iron ball would need to be approximately 455 °C in order to radiate a net power of 500 W into the surrounding air.

Question

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 equal to the Stefan-Boltzmann constant times the surface area of the object times the temperature to the fourth power. The equation is:

P = σ * A * T^4

where:
P is the power radiated,
σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W/m^2K^4),
A is the surface area of the object, and
T is the temperature in Kelvin.

We are given that P = 500 W and we need to find T.

First, we need to calculate the surface area of the iron ball. The surface area of a sphere is given by the equation A = 4πr^2. The radius r is given as 10 cm, but we need to convert this to meters, so r = 0.1 m.

A = 4π(0.1 m)^2 = 0.04π m^2

Now we can substitute P, σ, and A into the Stefan-Boltzmann Law and solve for T:

500 W = (5.67 * 10^-8 W/m^2K^4) * (0.04π m^2) * T^4

Solving for T gives:

T^4 = 500 W / [(5.67 * 10^-8 W/m^2K^4) * (0.04π m^2)]

T^4 = 2.21 * 10^9 K^4

Taking the fourth root of both sides gives:

T = (2.21 * 10^9 K^4)^(1/4) = 728 K

However, this is the temperature in Kelvin. To convert to Celsius, we subtract 273.15:

T = 728 K - 273.15 = 454.85 °C

So, the iron ball would need to be approximately 455 °C in order to radiate a net power of 500 W into the surrounding air.

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