A cylindrical-shaped object with a 3.4 cm radius and 8.9 cm height has an emissivity of 0.9. The object is heated up to 384 0C and then placed in a room with a surrounding temperature of 27 0CUsing the Stefan-Boltzmann law, calculate the magnitude of the net power emitted by the object through radiation.Give your answer rounded to 4 decimal places without units.

Question

BlackTom AI · Accepted Answer

Using the Stefan–Boltzmann law for net radiative power: P_net = ε σ A (T^4 − T_env^4).

1) Relevant concepts
- Emissivity ε scales the ideal blackbody emission.
- Temperatures must be in kelvin: T(K) = T(°C) + 273.15.
- Surface area A of a cylinder includes lateral area plus both circular ends: A = 2π r h + 2π r^2.
- Stefan–Boltzmann constant σ = 5.670374419 × 10^−8 W·m^−2·K^−4.

2) Step-by-step calculations
- Convert dimensions to meters: r = 3.4 cm = 0.034 m, h = 8.9 cm = 0.089 m.
- Convert temperatures to kelvin:
  T = 384 °C = 384 + 273.15 = 657.15 K
  T_env = 27 °C = 27 + 273.15 = 300.15 K
- Cylinder surface area:
  Lateral area = 2π r h = 2π(0.034)(0.089) = 6.283185307·0.003026 ≈ 0.01901291928 m^2
  End area = 2π r^2 = 2π(0.034)^2 = 6.283185307·0.001156 ≈ 0.00726336222 m^2
  Total A = 0.01901291928 + 0.00726336222 = 0.02627628150 m^2
- Compute T^2 values:
  T^2 = 657.15^2 = 431,846.1225 K^2
  T_env^2 = 300.15^2 = 90,090.0225 K^2
- Use (T^4 − T_env^4) = (T^2 − T_env^2)(T^2 + T_env^2):
  T^2 − T_env^2 = 431,846.1225 − 90,090.0225 = 341,756.10
  T^2 + T_env^2 = 431,846.1225 + 90,090.0225 = 521,936.145
  ⇒ T^4 − T_env^4 = 341,756.10 × 521,936.145 ≈ 1.783748613642345 × 10^11 K^4
- Now compute P_net:
  P_net = 0.9 × (5.670374419×10^−8) × 0.02627628150 × 1.783748613642345×10^11
  First, 0.9·σ·A ≈ 0.9 × 5.670374419e−8 × 0.02627628150 ≈ 1.340970×10^−9
  Multiply: P_net ≈ 1.340970×10^−9 × 1.783748613642345×10^11 ≈ 239.1948

3) Why this answer is correct
All steps follow the Stefan–Boltzmann law with correct area and Kelvin conversions; the intermediate arithmetic leads directly to 239.1948.

4) Other options
(Not applicable — this was a numerical question with a single computed result.)

Final confirmation: the net radiated power (rounded to four decimal places, no units) is 239.1948.

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