Which one of the following graphs shows the variation of magnetic induction B which distance r from a long wire carrying a current ?

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph shows the variation of magnetic induction \( B \) with distance \( r \) from a long wire carrying a current, we can follow these steps:

### Step 1: Understand the relationship between magnetic induction and distance
The magnetic induction \( B \) around a long straight current-carrying wire is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where:
- \( B \) is the magnetic induction,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current through the wire,
- \( r \) is the distance from the wire.

### Step 2: Analyze the formula
From the formula, we can see that \( B \) is inversely proportional to \( r \). This means that as the distance \( r \) from the wire increases, the magnetic induction \( B \) decreases.

### Step 3: Determine the nature of the graph
Since \( B \) decreases as \( r \) increases, the graph of \( B \) versus \( r \) will show a hyperbolic decay. Specifically, it will start at a maximum value when \( r \) is very small and will approach zero as \( r \) becomes very large.

### Step 4: Identify the correct graph
Among the options provided, we need to look for a graph that shows:
- A decreasing curve,
- Approaching zero as \( r \) increases,
- Not crossing the axis (since \( B \) cannot be negative).

### Conclusion
Based on the analysis, the correct option that represents the variation of magnetic induction \( B \) with distance \( r \) from a long wire carrying a current is **Option C**.

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