A solenoid is 2 m long and 4 cm in diameter. It has 4 layers of windings of 1000 turns each and carries a current of 5 A. What is the magnetic field at the center of the solenoid?

Options

A. `10^(-3)T`
B. `2pi xx 10^(-3)T`
C. `4pi xx 10^(-3)T`
D. `8pi xx 10^(-3)T`

Explanation

To find the magnetic field at the center of the solenoid, we can use the formula for the magnetic field 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,
- \( I \) is the current flowing through the solenoid.

### Step 1: Calculate the total number of turns
The solenoid has 4 layers of windings, with each layer having 1000 turns. Therefore, the total number of turns \( N \) is:

\[ N = 4 \times 1000 = 4000 \, \text{turns} \]

### Step 2: Calculate the length of the solenoid in meters
The length of the solenoid is given as 2 meters.

### Step 3: Calculate the number of turns per unit length \( n \)
The number of turns per unit length \( n \) can be calculated as:

\[ n = \frac{N}{L} = \frac{4000 \, \text{turns}}{2 \, \text{m}} = 2000 \, \text{turns/m} \]

### Step 4: Substitute values into the magnetic field formula
Now, we can substitute the values of \( \mu_0 \), \( n \), and \( I \) into the formula for the magnetic field:

\[ B = \mu_0 n I \]
\[ B = (4\pi \times 10^{-7} \, \text{T m/A}) \times (2000 \, \text{turns/m}) \times (5 \, \text{A}) \]

### Step 5: Perform the calculation
Calculating \( B \):

\[ B = 4\pi \times 10^{-7} \times 2000 \times 5 \]

Calculating the numerical values:

\[ B = 4\pi \times 10^{-7} \times 10000 \]
\[ B = 4\pi \times 10^{-3} \, \text{T} \]

### Final Answer
Thus, the magnetic field at the center of the solenoid is:

\[ B = 4\pi \times 10^{-3} \, \text{T} \]