A current `i` ampere flows along the inner conductor of a coaxial cable and returns along the outer conductor of the cable, then the magnetic induction at any point outside the conductor at a distance `r` meter from the axis is

Options

A. `oo`
B. Zero
C. `(mu_(0))/(4pi) (2i)/r`
D. `(mu_(0))/(4pi) (2pir)/r`

Explanation

To solve the problem of finding the magnetic induction (magnetic field) at a point outside a coaxial cable with a current `i` flowing in the inner conductor and returning in the outer conductor, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a coaxial cable consisting of an inner conductor and an outer conductor.
- The current `i` flows along the inner conductor and returns along the outer conductor.

2. **Identifying the Point of Interest**:
- We are interested in finding the magnetic field at a distance `r` from the axis of the coaxial cable, where `r` is greater than the radius of the outer conductor.

3. **Applying Ampere's Law**:
- According to Ampere's Law, the line integral of the magnetic field `B` around a closed loop is equal to the permeability of free space `μ₀` multiplied by the total current `I_enc` enclosed by that loop:
\[ \oint B \cdot dl = μ₀ I_{enc} \]

4. **Choosing an Amperian Loop**:
- We choose a circular Amperian loop of radius `r` centered on the axis of the coaxial cable. This loop is outside both the inner and outer conductors.

5. **Calculating the Enclosed Current**:
- Since the current `i` flows in the inner conductor and returns in the outer conductor, the total current enclosed by our Amperian loop is:
\[ I_{enc} = i - i = 0 \]
- Thus, the net current enclosed by the loop is zero.

6. **Applying Ampere's Law**:
- Substituting `I_enc = 0` into Ampere's Law gives:
\[ \oint B \cdot dl = μ₀ \cdot 0 = 0 \]
- This implies that the magnetic field `B` at distance `r` is zero.

7. **Conclusion**:
- Therefore, the magnetic induction (magnetic field) at any point outside the coaxial cable is:
\[ B = 0 \]

### Final Answer:
The magnetic induction at any point outside the conductor at a distance `r` from the axis is **0**.

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