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 regarding the variation of the magnetic field with distance from the axis in a coaxial cable with two conductors carrying equal currents in opposite directions, we can follow these steps:

### Step-by-Step Solution:

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

2. **Apply Ampère's Law**:
- To find the magnetic field at a distance \( r \) from the axis, we can use Ampère's Law, which states that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is equal to the permeability of free space times the total current \( I_{enc} \) enclosed by the loop:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \]

3. **Consider Different Regions**:
- **Region 1**: Inside the inner conductor (\( r < R_1 \)):
- The magnetic field \( B = 0 \) because there is no current enclosed.
- **Region 2**: Between the inner and outer conductors (\( R_1 < r < R_2 \)):
- The enclosed current is \( I \). Thus, applying Ampère's Law:
\[ B(2\pi r) = \mu_0 I \implies B = \frac{\mu_0 I}{2\pi r} \]
- **Region 3**: Inside the outer conductor (\( R_2 < r < R_3 \)):
- The enclosed current is \( I - I = 0 \). Thus, \( B = 0 \).
- **Region 4**: Outside the outer conductor (\( r > R_3 \)):
- The total enclosed current is \( I - I = 0 \). Thus, \( B = 0 \).

4. **Summarize the Magnetic Field Variation**:
- The magnetic field \( B \) varies with distance \( r \) as follows:
- \( B = 0 \) for \( r < R_1 \)
- \( B = \frac{\mu_0 I}{2\pi r} \) for \( R_1 < r < R_2 \)
- \( B = 0 \) for \( R_2 < r < R_3 \)
- \( B = 0 \) for \( r > R_3 \)

5. **Plot the Magnetic Field**:
- The graph of \( B \) versus \( r \) will show:
- A value of \( 0 \) for \( r < R_1 \)
- A hyperbolic decrease from \( \frac{\mu_0 I}{2\pi R_1} \) to \( 0 \) as \( r \) approaches \( R_2 \)
- A value of \( 0 \) for \( R_2 < r < R_3 \)
- A value of \( 0 \) for \( r > R_3 \)

### Final Answer:
The variation of the magnetic field with distance from the axis is best plotted as a hyperbolic curve between \( R_1 \) and \( R_2 \), and zero elsewhere.