A solenoid of length 50 cm and a radius of cros section 1 cm has 1000 turns of wire wound over it if the current carried is 5A the magnetic field on its axis near the centre of the solenoid is approximately (Given permeability of free space `mu_(0)=4pi xx10^(-7) T-mA^(-1)`)

Options

A. `0.63xx10^(-2)T`
B. `1.26 xx10^(-2) T`
C. `2.51 xx10^(-2)T`
D. `6.3 T`

Explanation

To solve the problem of finding the magnetic field on the axis of a solenoid, we can use the formula for the magnetic field inside a long solenoid, which is given by:

\[ B = \mu_0 \cdot n \cdot I \]

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 of the solenoid,
- \( I \) is the current flowing through the solenoid.

### Step-by-Step Solution:

1. **Identify the given values:**
- Length of the solenoid, \( L = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Radius of cross-section, \( r = 1 \, \text{cm} \) (not needed for this calculation)
- Number of turns, \( N = 1000 \)
- Current, \( I = 5 \, \text{A} \)
- Permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)

2. **Calculate the number of turns per unit length (n):**
\[ n = \frac{N}{L} = \frac{1000 \, \text{turns}}{0.5 \, \text{m}} = 2000 \, \text{turns/m} \]

3. **Substitute the values into the magnetic field formula:**
\[ B = \mu_0 \cdot n \cdot I \]
\[ B = (4\pi \times 10^{-7} \, \text{T m/A}) \cdot (2000 \, \text{turns/m}) \cdot (5 \, \text{A}) \]

4. **Calculate the magnetic field:**
\[ B = 4\pi \times 10^{-7} \cdot 2000 \cdot 5 \]
\[ B = 4\pi \times 10^{-7} \cdot 10000 \]
\[ B = 4\pi \times 10^{-3} \, \text{T} \]

5. **Use the value of \(\pi \approx 3.14\) to find the final answer:**
\[ B \approx 4 \cdot 3.14 \times 10^{-3} \, \text{T} = 12.56 \times 10^{-3} \, \text{T} = 1.256 \times 10^{-2} \, \text{T} \]

### Final Answer:
The magnetic field on the axis near the center of the solenoid is approximately \( 1.256 \times 10^{-2} \, \text{T} \) or \( 12.56 \, \text{mT} \).