A solenoid of length `0.250` m and radius `0.0200` m is comprised of `120` turns of wire. Determine the magnitude of the magnetic field at the center of the solenoid when it carries a current of `15.0` A.

Options

A. `2.26xx10^(-3)T`
B. `9.05xx10^(-3)T`
C. `4.52xx10^(-3)T`
D. `7.50xx10^(-3)T`

Answer

To determine the magnitude of the magnetic field at the center of the solenoid, we can use the formula for the magnetic field inside a long solenoid, which is given by:

\[ B = \mu_0 \frac{N I}{L} \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( N \) is the number of turns of the solenoid,
- \( I \) is the current flowing through the solenoid,
- \( L \) is the length of the solenoid.

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of the solenoid, \( L = 0.250 \, \text{m} \)
- Radius of the solenoid, \( r = 0.0200 \, \text{m} \) (not needed for the calculation)
- Number of turns, \( N = 120 \)
- Current, \( I = 15.0 \, \text{A} \)

2. **Substitute the values into the formula:**
- Using the permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \),
- Substitute \( N \), \( I \), and \( L \) into the formula:

\[ B = \left(4\pi \times 10^{-7} \, \text{T m/A}\right) \frac{120 \, \text{turns} \times 15.0 \, \text{A}}{0.250 \, \text{m}} \]

3. **Calculate the numerator:**
- Calculate \( 120 \times 15.0 = 1800 \).

4. **Calculate the entire expression:**
- Now substitute back into the equation:

\[ B = \left(4\pi \times 10^{-7}\right) \frac{1800}{0.250} \]

- Calculate \( \frac{1800}{0.250} = 7200 \).

5. **Final calculation:**
- Now calculate \( B \):

\[ B = 4\pi \times 10^{-7} \times 7200 \]

- Using \( \pi \approx 3.14 \):

\[ B \approx 4 \times 3.14 \times 10^{-7} \times 7200 \]

- Calculate \( 4 \times 3.14 \approx 12.56 \):

\[ B \approx 12.56 \times 10^{-7} \times 7200 \]

- Finally, calculate \( 12.56 \times 7200 \approx 90500 \).

- Therefore,

\[ B \approx 9.05 \times 10^{-3} \, \text{T} \]

### Conclusion:
The magnitude of the magnetic field at the center of the solenoid is approximately \( 9.05 \, \text{mT} \) (milliTesla).