A solenoid has length `0.4m`, radius 1 cm and 400 turns of wire. If a current fo 5 A is passed through this solenoid, then what is the magnetic field inside the solenoid?

Options

A. `6.28xx10^(-4)T`
B. `6.28xx10^(-3)T`
C. `6.28xx10^(-7)T`
D. `6.28xx10^(-6)T`

Answer

To find the magnetic field inside a solenoid, we can use the formula:

\[ B = \mu_0 n I \]

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

### Step 1: Identify the given values
- Length of the solenoid, \( L = 0.4 \, \text{m} \)
- Radius of the solenoid, \( r = 1 \, \text{cm} = 0.01 \, \text{m} \) (not needed for the calculation of \( B \))
- Total number of turns, \( N = 400 \)
- Current, \( I = 5 \, \text{A} \)

### Step 2: Calculate the number of turns per unit length (\( n \))
The number of turns per unit length is given by:

\[ n = \frac{N}{L} \]

Substituting the values:

\[ n = \frac{400}{0.4} = 1000 \, \text{turns/m} \]

### Step 3: Substitute values into the magnetic field formula
Now we can substitute \( n \) and \( I \) into the formula for \( B \):

\[ B = \mu_0 n I \]

Substituting the values:

\[ B = (4\pi \times 10^{-7} \, \text{T m/A}) \times (1000 \, \text{turns/m}) \times (5 \, \text{A}) \]

### Step 4: Calculate \( B \)
Calculating the above expression:

\[ B = 4\pi \times 10^{-7} \times 1000 \times 5 \]

\[ B = 20\pi \times 10^{-4} \, \text{T} \]

Using \( \pi \approx 3.14 \):

\[ B \approx 20 \times 3.14 \times 10^{-4} \, \text{T} \approx 62.8 \times 10^{-4} \, \text{T} = 6.28 \times 10^{-3} \, \text{T} \]

### Final Answer
Thus, the magnetic field inside the solenoid is:

\[ B \approx 6.28 \times 10^{-3} \, \text{T} \]

---