Which of the following figure shown the magnetic flux denstiy b at a distance r from a long straight rod carrying a steady current I ?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining which figure shows the magnetic flux density \( B \) at a distance \( r \) from a long straight rod carrying a steady current \( I \), we can follow these steps:

### Step 1: Understand the Magnetic Field Around a Long Straight Current-Carrying Rod
The magnetic field \( B \) around a long straight rod carrying a steady current can be derived using Ampere's Circuital Law. The law states that the line integral of the magnetic field \( B \) around a closed loop is equal to the permeability of free space \( \mu_0 \) times the total current \( I \) enclosed by the loop.

### Step 2: Analyze the Two Cases
We need to analyze two cases based on the distance \( r \) from the center of the rod:

1. **Case 1: When \( r \) is greater than the radius \( A \) of the rod.**
- Here, the magnetic field \( B \) can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- This shows that \( B \) is inversely proportional to \( r \) (i.e., as \( r \) increases, \( B \) decreases).

2. **Case 2: When \( r \) is less than the radius \( A \) of the rod.**
- In this case, the current enclosed by the Amperian loop is not the total current \( I \) but a fraction of it based on the area. The enclosed current \( I_{\text{enclosed}} \) can be expressed as:
\[ I_{\text{enclosed}} = I \left( \frac{r^2}{A^2} \right) \]
- The magnetic field \( B \) in this case is given by:
\[ B = \frac{\mu_0 I}{2 \pi A^2} r \]
- This indicates that \( B \) is directly proportional to \( r \) (i.e., as \( r \) increases, \( B \) increases).

### Step 3: Graphical Representation
To represent these relationships graphically:
- For \( r < A \), the graph will be a straight line passing through the origin, indicating \( B \) is directly proportional to \( r \).
- For \( r > A \), the graph will show a hyperbolic decline, indicating \( B \) is inversely proportional to \( r \).

### Step 4: Identify the Correct Option
Based on the analysis:
- The graph will have a linear portion for \( r < A \) and a hyperbolic portion for \( r > A \).
- The correct option that represents this behavior is **Option D**.

### Final Answer
The answer to the question is **Option D**.

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