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` of electric power is needed to do it. When the surface of the copper sphere is completely blackened, `700W` is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

Explanation

To solve the problem, we will use the concept of thermal radiation and the Stefan-Boltzmann law. The power radiated by a body is given by:

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

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

### Step 1: Set up equations for both scenarios

1. **For the copper sphere (not blackened)**:
- Given that \( P_1 = 210 \, \text{W} \), \( T = 500 \, \text{K} \), and \( T_0 = 300 \, \text{K} \):
\[ 210 = \varepsilon \sigma A (500^4 - 300^4) \tag{1} \]

2. **For the blackened copper sphere**:
- Given that \( P_2 = 700 \, \text{W} \):
\[ 700 = 1 \cdot \sigma A (500^4 - 300^4) \tag{2} \]
(Here, emissivity \( \varepsilon = 1 \) for a black body)

### Step 2: Calculate \( 500^4 - 300^4 \)

We need to calculate \( 500^4 - 300^4 \):
\[ 500^4 = 62500000000 \quad \text{and} \quad 300^4 = 8100000000 \]
\[ 500^4 - 300^4 = 62500000000 - 8100000000 = 54499999999 \]

### Step 3: Substitute the value into equations

Substituting this value into both equations:

1. From equation (1):
\[ 210 = \varepsilon \sigma A (54499999999) \]

2. From equation (2):
\[ 700 = \sigma A (54499999999) \]

### Step 4: Solve for \( \varepsilon \)

Now, we can divide equation (1) by equation (2):
\[ \frac{210}{700} = \frac{\varepsilon \sigma A (54499999999)}{\sigma A (54499999999)} \]
This simplifies to:
\[ \frac{3}{10} = \varepsilon \]

Thus, the emissivity \( \varepsilon \) of copper is:
\[ \varepsilon = 0.3 \]

### Final Answer

The emissivity of copper is **0.3**.

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