Five hundred turns of a wire are wound oa, a thin tubc 1m Iong. If the wire carries a current of 5A , determine the field (in mT) in the tube.

Answer

To determine the magnetic field inside a solenoid, we can use the formula for the magnetic field \( B \) inside a solenoid, which is given by:

\[ B = \mu_0 n I \]

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 per unit length (turns/m),
- \( I \) is the current in amperes (A).

### Step 1: Calculate the number of turns per unit length \( n \)

Given:
- Total number of turns \( N = 500 \)
- Length of the solenoid \( L = 1 \, \text{m} \)

The number of turns per unit length \( n \) can be calculated as:

\[ n = \frac{N}{L} = \frac{500}{1} = 500 \, \text{turns/m} \]

### Step 2: Substitute values into the magnetic field formula

Now, we will substitute the values into the formula for the magnetic field \( B \):

\[ B = \mu_0 n I \]

Substituting \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \), \( n = 500 \, \text{turns/m} \), and \( I = 5 \, \text{A} \):

\[ B = (4\pi \times 10^{-7}) \times (500) \times (5) \]

### Step 3: Calculate the magnetic field \( B \)

Now, we will calculate the value step by step:

1. Calculate \( 500 \times 5 = 2500 \).
2. Now multiply \( 4\pi \times 10^{-7} \) by \( 2500 \):

\[ B = 4\pi \times 10^{-7} \times 2500 \]

Using \( \pi \approx 3.14 \):

\[ B \approx 4 \times 3.14 \times 10^{-7} \times 2500 \]

Calculating \( 4 \times 3.14 = 12.56 \):

\[ B \approx 12.56 \times 10^{-7} \times 2500 \]

Calculating \( 12.56 \times 2500 = 31400 \):

\[ B \approx 31400 \times 10^{-7} = 3.14 \times 10^{-3} \, \text{T} \]

### Step 4: Convert to millitesla (mT)

Since \( 1 \, \text{T} = 1000 \, \text{mT} \):

\[ B \approx 3.14 \times 10^{-3} \, \text{T} = 3.14 \, \text{mT} \]

### Final Answer

The magnetic field in the tube is approximately:

\[ \boxed{3.14 \, \text{mT}} \]

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