A long solid cylinder is radiating power. It is remolded into a number of smaller cylinders, each of which has the same length as original cylinder. Each small cylinder has the same temperature as the original cylinder. The total radiant power emitted by the pieces is twice that emitted by the original cylinder. How many smaller cylinders are there? (Neglect the energy emitted by the flat faces of cylinder.)

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Given Information
We have a long solid cylinder that is remolded into several smaller cylinders. Each smaller cylinder has the same length as the original cylinder and the same temperature. The total radiant power emitted by the smaller cylinders is twice that emitted by the original cylinder.

### Step 2: Use the Formula for Radiant Power
The power radiated by a surface is given by the Stefan-Boltzmann Law:
\[ P = \sigma E A T^4 \]
where:
- \( P \) is the power emitted,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( E \) is the emissivity of the material,
- \( A \) is the surface area,
- \( T \) is the absolute temperature.

### Step 3: Calculate the Area of the Original Cylinder
For the original cylinder:
- The surface area \( A \) (neglecting the flat faces) is given by:
\[ A = 2 \pi R L \]
where \( R \) is the radius and \( L \) is the length of the cylinder.

Thus, the power emitted by the original cylinder is:
\[ P = \sigma E (2 \pi R L) T^4 \]

### Step 4: Calculate the Area of the Smaller Cylinders
Let the number of smaller cylinders be \( n \) and the radius of each smaller cylinder be \( r \). Since the volume of the original cylinder must equal the total volume of the smaller cylinders, we have:
\[ \pi R^2 L = n \cdot \pi r^2 L \]
This simplifies to:
\[ R^2 = n r^2 \]
or
\[ r = \frac{R}{\sqrt{n}} \]

### Step 5: Calculate the Power Emitted by One Smaller Cylinder
The surface area of one smaller cylinder is:
\[ A' = 2 \pi r L = 2 \pi \left(\frac{R}{\sqrt{n}}\right) L \]
Thus, the power emitted by one smaller cylinder is:
\[ P' = \sigma E (2 \pi r L) T^4 = \sigma E \left(2 \pi \left(\frac{R}{\sqrt{n}}\right) L\right) T^4 \]

### Step 6: Calculate the Total Power Emitted by All Smaller Cylinders
The total power emitted by all \( n \) smaller cylinders is:
\[ P_{\text{total}} = n P' = n \cdot \sigma E \left(2 \pi \left(\frac{R}{\sqrt{n}}\right) L\right) T^4 \]
This simplifies to:
\[ P_{\text{total}} = \sigma E (2 \pi R L) T^4 \cdot \sqrt{n} \]

### Step 7: Set Up the Equation Based on Given Conditions
According to the problem, the total power emitted by the smaller cylinders is twice that of the original cylinder:
\[ P_{\text{total}} = 2P \]
Substituting the expressions we derived:
\[ \sigma E (2 \pi R L) T^4 \cdot \sqrt{n} = 2 \cdot \sigma E (2 \pi R L) T^4 \]

### Step 8: Cancel Common Terms
We can cancel \( \sigma E (2 \pi R L) T^4 \) from both sides:
\[ \sqrt{n} = 2 \]

### Step 9: Solve for \( n \)
Squaring both sides gives:
\[ n = 4 \]

### Conclusion
Thus, the number of smaller cylinders is \( n = 4 \).

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