A copper sphere is suspended in an evacuated chamber maintained at 300K. The sphere is maitained 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 blackned, 700W is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

Explanation

To solve the problem of finding the emissivity of the copper sphere, we can follow these steps:

### Step 1: Understand the Problem
We have a copper sphere at a constant temperature of 500 K in an evacuated chamber at 300 K. The power required to maintain this temperature when the sphere is in its natural state is 210 W. When the sphere is blackened, the power required increases to 700 W. We need to find the emissivity of the copper sphere.

### Step 2: Use the Stefan-Boltzmann Law
According to the Stefan-Boltzmann Law, the power emitted by a body is given by:
\[ P = \epsilon \sigma A (T_1^4 - T_0^4) \]
where:
- \( P \) is the power emitted,
- \( \epsilon \) is the emissivity,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T_1 \) is the temperature of the body (500 K),
- \( T_0 \) is the temperature of the surroundings (300 K).

### Step 3: Set Up the Equations
For the copper sphere when it is not blackened, the power equation can be written as:
\[ P_c = \epsilon_c \sigma A (T_1^4 - T_0^4) \]
Given \( P_c = 210 \, W \), we can write:
\[ 210 = \epsilon_c \sigma A (500^4 - 300^4) \] (Equation 1)

For the blackened sphere (acting as a black body), the emissivity is 1:
\[ P_b = \sigma A (T_1^4 - T_0^4) \]
Given \( P_b = 700 \, W \), we can write:
\[ 700 = \sigma A (500^4 - 300^4) \] (Equation 2)

### Step 4: Divide the Equations
From Equation 1 and Equation 2, we can eliminate \( \sigma A (500^4 - 300^4) \):
\[ \frac{210}{\epsilon_c} = 700 \]
Rearranging gives:
\[ \epsilon_c = \frac{210}{700} \]

### Step 5: Calculate Emissivity
Calculating the above expression:
\[ \epsilon_c = \frac{210}{700} = 0.3 \]

### Final Answer
The emissivity of the copper sphere is \( \epsilon_c = 0.3 \).

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