If the absolute temperature of a body is doubled, the power radiated will increase by a factor of

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem of how the power radiated by a body changes when its absolute temperature is doubled, we can use Stefan-Boltzmann's Law. Here’s a step-by-step solution:

### Step 1: Understand Stefan-Boltzmann's Law
Stefan-Boltzmann's Law states that the power radiated by a black body is directly proportional to the fourth power of its absolute temperature (T). Mathematically, this can be expressed as:
\[ P \propto T^4 \]
where \( P \) is the power radiated and \( T \) is the absolute temperature.

### Step 2: Set Up the Initial and Final Conditions
Let:
- The initial temperature be \( T_1 = T \)
- The initial power radiated be \( P_1 = P \)

When the temperature is doubled:
- The final temperature will be \( T_2 = 2T \)
- The final power radiated will be \( P_2 = P' \)

### Step 3: Apply Stefan-Boltzmann's Law
According to the law, we can write:
\[ P_1 = k T_1^4 \]
\[ P_2 = k T_2^4 \]
where \( k \) is a constant of proportionality.

### Step 4: Substitute the Values
Now substituting the values:
\[ P_1 = k T^4 \]
\[ P_2 = k (2T)^4 \]

Calculating \( P_2 \):
\[ P_2 = k (2T)^4 = k \cdot 16T^4 \]

### Step 5: Relate the Powers
Now we can relate the two powers:
\[ P' = 16 P \]

### Step 6: Determine the Factor of Increase
To find the factor by which the power has increased:
\[ \text{Factor of increase} = \frac{P'}{P} = \frac{16P}{P} = 16 \]

Thus, if the absolute temperature of a body is doubled, the power radiated will increase by a factor of **16**.

### Final Answer
The power radiated will increase by a factor of **16**.

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