To find the radius of the spherical body using the Stefan-Boltzmann Law, we first need to understand how this law relates the power radiated by a black body to its temperature. The law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. The formula is given by:
Stefan-Boltzmann Law
The Stefan-Boltzmann Law can be expressed mathematically as:
P = εσAT^4
- P = power radiated (in watts)
- ε = emissivity of the material (dimensionless)
- σ = Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/m²K^4)
- A = surface area of the sphere (in m²)
- T = absolute temperature of the body (in Kelvin)
Given Values
From the problem, we know:
- Emissivity (ε) = 0.250
- Power input (P) = 5.5 × 10^3 W
- Temperature of the sphere (T) = 2500 K
Calculating the Surface Area
The surface area (A) of a sphere is calculated using the formula:
A = 4πr²
Where r is the radius of the sphere. We can substitute this into the Stefan-Boltzmann equation:
P = εσ(4πr²)T^4
Rearranging the Equation
Now, we can rearrange this equation to solve for the radius r:
r² = P / (εσ(4πT^4))
r = √(P / (εσ(4πT^4)))
Substituting the Values
Now, let’s plug in the values:
- ε = 0.250
- σ = 5.67 × 10^-8 W/m²K^4
- T = 2500 K
- P = 5.5 × 10^3 W
Substituting these into the equation:
r = √(5.5 × 10^3 / (0.250 × 5.67 × 10^-8 × 4π × (2500)^4))
Calculating Step-by-Step
First, calculate the term inside the square root:
- Calculate T^4: (2500 K)⁴ = 3.90625 × 10^13 K⁴
- Calculate 4πT^4: 4π × 3.90625 × 10^13 ≈ 4.898 × 10^14
- Calculate εσ(4πT^4): 0.250 × 5.67 × 10^-8 × 4.898 × 10^14 ≈ 0.250 × 2.653 × 10^7 ≈ 6.6325 × 10^6
Now, substitute this back into the equation for r:
r = √(5.5 × 10^3 / 6.6325 × 10^6)
Calculating this gives:
r ≈ √(0.000829) ≈ 0.0288 m
Final Result
The radius of the spherical body is approximately 0.0288 meters or 28.8 mm.
