A copper sphere is suspended in an evacuated chamber maintained at 300K. The sphere is maintained at a constant temperature of 500K by heating it electrically. A total of 210W is electric power is needed to do it. When the surface of the copper sphere is completely blanked, 700W is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

Explanation

To find the emissivity of the copper sphere, we can use Stefan-Boltzmann's law, which relates the power radiated by a black body to its temperature. The formula is given by:

\[ P = e \cdot A \cdot \sigma \cdot (T^4 - T_0^4) \]

where:
- \( P \) is the power,
- \( e \) is the emissivity,
- \( A \) is the surface area,
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4 \)),
- \( T \) is the temperature of the object in Kelvin,
- \( T_0 \) is the ambient temperature in Kelvin.

### Step 1: Write the equations for both scenarios

1. **For the copper sphere at 500 K with power 210 W:**
\[ 210 = e \cdot A \cdot \sigma \cdot (500^4 - 300^4) \quad \text{(Equation 1)} \]

2. **For the blackened surface at 500 K with power 700 W:**
\[ 700 = 1 \cdot A \cdot \sigma \cdot (500^4 - 300^4) \quad \text{(Equation 2)} \]

### Step 2: Simplify the equations

From Equation 2, we can express the area \( A \cdot \sigma \cdot (500^4 - 300^4) \):
\[ A \cdot \sigma \cdot (500^4 - 300^4) = 700 \]

### Step 3: Substitute Equation 2 into Equation 1

Now substitute the expression for \( A \cdot \sigma \cdot (500^4 - 300^4) \) from Equation 2 into Equation 1:
\[ 210 = e \cdot 700 \]

### Step 4: Solve for emissivity \( e \)

Now, isolate \( e \):
\[ e = \frac{210}{700} \]
\[ e = 0.3 \]

### Conclusion

The emissivity of copper is \( 0.3 \).

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