The correct curve between the magnetic induction (B) along the axis of a along solenoid due to current flow i in it and distance x from one end is -

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct curve between the magnetic induction (B) along the axis of a long solenoid due to current flow (I) and the distance (x) from one end, we can follow these steps:

### Step 1: Understand the Magnetic Field in a Long Solenoid
The magnetic field (B) inside a long solenoid can be expressed as:
\[ B = \mu_0 n I \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length of the solenoid,
- \( I \) is the current flowing through the solenoid.

### Step 2: Analyze the Magnetic Field at Different Points
1. **At one end of the solenoid (x = 0)**:
- The magnetic field is given by:
\[ B = \frac{\mu_0 n I}{2} \]
(This is because at the end of the solenoid, the angle \( \theta_1 \) is 90 degrees and \( \theta_2 \) is 0 degrees).

2. **At the center of the solenoid**:
- The magnetic field is maximum and is given by:
\[ B = \mu_0 n I \]
(At the center, both angles \( \theta_1 \) and \( \theta_2 \) are close to 0 degrees).

3. **At the other end of the solenoid (x = L)**:
- The magnetic field is again:
\[ B = \frac{\mu_0 n I}{2} \]
(Here, \( \theta_1 \) is 0 degrees and \( \theta_2 \) is 90 degrees).

### Step 3: Sketch the Graph
- The magnetic field starts at \( \frac{\mu_0 n I}{2} \) at one end (x = 0), increases to \( \mu_0 n I \) at the center (x = L/2), and then decreases back to \( \frac{\mu_0 n I}{2} \) at the other end (x = L).
- Therefore, the graph of B versus x will be a curve that rises to a maximum at the center and then falls back down, resembling a bell-shaped curve.

### Step 4: Identify the Correct Option
- Based on the analysis, the correct curve that represents the relationship between magnetic induction (B) and distance (x) from one end of the solenoid is the one that starts at \( \frac{\mu_0 n I}{2} \), reaches a maximum of \( \mu_0 n I \) at the center, and returns to \( \frac{\mu_0 n I}{2} \) at the other end.

### Conclusion
The correct option is the one that shows this bell-shaped curve, indicating that the magnetic field increases from one end to the center and then decreases towards the other end.

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