A coaxial cable made up of two conductors. The inner conductor is solid and is of radius `R_1` and the outer conductor is hollow of inner radius `R_2` and outer radius `R_3`. The space between the conductors is filled with air. The inner and outer conductors are carrying currents of equal magnitudes and in opposite directions. Then the variation of magnetic field with distance from the axis is best plotted as

Options

A.
B.
C.
D.

Answer

To solve the problem of the magnetic field variation in a coaxial cable with two conductors carrying equal but opposite currents, we can follow these steps:

### Step 1: Understand the Configuration
We have a coaxial cable with:
- An inner solid conductor of radius \( R_1 \).
- An outer hollow conductor with inner radius \( R_2 \) and outer radius \( R_3 \).
- The inner and outer conductors carry equal currents in opposite directions.

### Step 2: Apply Ampere's Law
According to Ampere's Law, the magnetic field \( B \) around a current-carrying conductor can be determined by the enclosed current. The law states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space times the current enclosed by that loop.

### Step 3: Analyze Different Regions
1. **Region 1: Inside the Inner Conductor (0 to \( R_1 \))**
- The magnetic field \( B \) is zero at the center (due to symmetry).
- As we move outward from the center to the surface of the inner conductor, the magnetic field increases linearly with distance from the axis.

2. **Region 2: Between the Inner and Outer Conductors (\( R_1 \) to \( R_2 \))**
- Here, we enclose the current from the inner conductor only.
- The magnetic field decreases as we move outward, but it does so in a hyperbolic manner due to the influence of the outer conductor's current.

3. **Region 3: Outside the Outer Conductor (\( R_2 \) to \( R_3 \))**
- In this region, the magnetic field continues to decrease hyperbolically, but the slope may differ due to the different media and the nature of the currents.

### Step 4: Plotting the Magnetic Field
- From \( 0 \) to \( R_1 \): The magnetic field \( B \) increases linearly.
- From \( R_1 \) to \( R_2 \): The magnetic field \( B \) decreases hyperbolically.
- From \( R_2 \) to \( R_3 \): The magnetic field \( B \) continues to decrease hyperbolically but at a different rate.

### Conclusion
The variation of the magnetic field with distance from the axis is best represented by a graph that shows:
- A linear increase from \( 0 \) to \( R_1 \).
- A hyperbolic decrease from \( R_1 \) to \( R_2 \).
- A continued hyperbolic decrease from \( R_2 \) to \( R_3 \).

Thus, the correct option is **C**.

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