The magnetic field on the axis of a long solenoid having n turns per unit length and carrying a current is

Options

A. `mu_(0)ni`
B. `mu_(0)n^(2)i`
C. `mu_(0)ni^(2)`
D. none of these

Explanation

To find the magnetic field on the axis of a long solenoid with \( n \) turns per unit length and carrying a current \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Solenoid**:
A solenoid is a long coil of wire that generates a magnetic field when an electric current passes through it. The magnetic field inside a long solenoid is uniform and parallel to the axis of the solenoid.

2. **Define the Parameters**:
- Let \( n \) be the number of turns per unit length of the solenoid.
- Let \( I \) be the current flowing through the solenoid.

3. **Applying Ampere's Circuital Law**:
According to Ampere's Circuital Law, the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is equal to \( \mu_0 \) times the total current \( I_{\text{enc}} \) passing through the loop:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

4. **Choosing an Amperian Loop**:
We choose a rectangular Amperian loop that runs inside the solenoid along its axis and extends outside it. The magnetic field inside the solenoid is uniform, while outside it is negligible.

5. **Calculating the Current Enclosed**:
The total current enclosed by the Amperian loop is:
\[ I_{\text{enc}} = n \cdot I \]
where \( n \) is the number of turns per unit length and \( I \) is the current through each turn.

6. **Evaluating the Integral**:
Inside the solenoid, the magnetic field \( B \) is uniform, and the length of the path inside the solenoid is \( l \). Thus, we can write:
\[ B \cdot l = \mu_0 (n \cdot I) \]
where \( l \) is the length of the solenoid.

7. **Solving for the Magnetic Field**:
Rearranging the equation gives us:
\[ B = \frac{\mu_0 n I}{l} \]
Since we are considering a long solenoid, we can simplify this to:
\[ B = \mu_0 n I \]

### Final Result:
The magnetic field on the axis of a long solenoid is given by:
\[ B = \mu_0 n I \]