To find the length of the solenoid, we can use the relationship between the magnetic moment, magnetization, and the dimensions of the solenoid. Here’s a step-by-step solution:
### Step 1: Understand the Given Data
- Radius of the solenoid, \( R = 4 \, \text{cm} = 0.04 \, \text{m} \)
- Number of turns, \( N = 1000 \)
- Current, \( I = 8 \, \text{A} \)
- Magnetization, \( M = 20000 \, \text{A/m} \)
### Step 2: Formula for Magnetic Moment
The magnetic moment \( \mu \) of the solenoid can be expressed as:
\[
\mu = N \cdot I \cdot A
\]
where \( A \) is the cross-sectional area of the solenoid.
### Step 3: Calculate the Cross-Sectional Area
The cross-sectional area \( A \) of the solenoid can be calculated using the formula for the area of a circle:
\[
A = \pi R^2 = \pi (0.04)^2 = \pi (0.0016) \approx 0.0050265 \, \text{m}^2
\]
### Step 4: Substitute into the Magnetic Moment Formula
Substituting the values into the magnetic moment formula:
\[
\mu = N \cdot I \cdot A = 1000 \cdot 8 \cdot 0.0050265 \approx 40.212 \, \text{A m}^2
\]
### Step 5: Relate Magnetic Moment to Magnetization
Magnetization \( M \) is defined as:
\[
M = \frac{\mu}{V}
\]
where \( V \) is the volume of the solenoid. The volume \( V \) can be expressed as:
\[
V = A \cdot L
\]
Thus, we can write:
\[
M = \frac{\mu}{A \cdot L}
\]
### Step 6: Rearranging for Length \( L \)
From the equation above, we can rearrange to find \( L \):
\[
L = \frac{\mu}{M \cdot A}
\]
### Step 7: Substitute the Known Values
Now, substituting the known values:
\[
L = \frac{40.212}{20000 \cdot 0.0050265}
\]
### Step 8: Calculate Length \( L \)
Calculating the denominator:
\[
20000 \cdot 0.0050265 \approx 100.53
\]
Now substituting this back:
\[
L = \frac{40.212}{100.53} \approx 0.399 \, \text{m}
\]
### Step 9: Convert to Centimeters
To convert meters to centimeters:
\[
L \approx 0.399 \, \text{m} \approx 39.9 \, \text{cm} \approx 40 \, \text{cm}
\]
### Final Answer
The length of the solenoid is approximately \( 40 \, \text{cm} \).
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