What will happen to the inductance of a solenoid if it is stretched to twice its length while maintaining the same number of turns and cross-sectional area?

Options

A. Its inductance will double
B. Its inductance will be reduced to half
C. Its inductance will remain unchanged
D. Its inductance will be reduced to one fourth

Answer

To determine what happens to the inductance of a solenoid when it is stretched to twice its length while maintaining the same number of turns and cross-sectional area, we can follow these steps:

### Step 1: Understand the Formula for Inductance
The inductance \( L \) of a solenoid is given by the formula:
\[ L = \frac{\mu_0 N I}{A L} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( I \) is the current flowing through the solenoid,
- \( A \) is the cross-sectional area,
- \( L \) is the length of the solenoid.

### Step 2: Identify Changes in Parameters
In this scenario, we are stretching the solenoid to twice its original length. Therefore, if the original length is \( L \), the new length \( L' \) will be:
\[ L' = 2L \]
The number of turns \( N \) and the cross-sectional area \( A \) remain unchanged.

### Step 3: Substitute into the Inductance Formula
Now we substitute the new length \( L' \) into the inductance formula:
\[ L' = \frac{\mu_0 N I}{A (2L)} \]

### Step 4: Simplify the Expression
We can simplify the expression for the new inductance \( L' \):
\[ L' = \frac{\mu_0 N I}{2AL} \]
This can be rewritten as:
\[ L' = \frac{1}{2} \left( \frac{\mu_0 N I}{A L} \right) \]
Thus, we see that:
\[ L' = \frac{L}{2} \]
where \( L \) is the original inductance.

### Step 5: Conclusion
From the simplification, we conclude that the inductance of the solenoid when stretched to twice its length is half of its original value:
\[ L' = \frac{L}{2} \]

### Final Answer
The inductance of the solenoid will be reduced to half of its original value.

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