In a coaxial, straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero.

Options

A. outside the cable
B. inside the inner conductor
C. inside the outer conductor
D. in between the two conductor

Answer

To solve the problem of finding the magnetic field in a coaxial straight cable with equal currents in opposite directions, we can follow these steps:

### Step 1: Understand the Configuration
We have a coaxial cable consisting of a central conductor and an outer conductor. The central conductor carries a current \( I \) in one direction, while the outer conductor carries an equal current \( I \) in the opposite direction.

### Step 2: Draw the Diagram
Draw a diagram to represent the coaxial cable. Label the central conductor and the outer conductor. Indicate the direction of the currents in both conductors. The central conductor can be represented with a dot (indicating current coming out of the page), and the outer conductor can be represented with an 'X' (indicating current going into the page).

### Step 3: Apply Ampere's Circuital Law
We will use Ampere's Circuital Law, which states:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{in}} \]
where \( I_{\text{in}} \) is the current enclosed by the Amperian loop.

### Step 4: Consider Different Regions
1. **Inside the Inner Conductor**: The magnetic field inside a solid conductor is determined by the current flowing through it. However, since we are looking for the magnetic field outside the outer conductor, we can skip this region.

2. **Between the Conductors**: In this region, the magnetic field due to the inner conductor is directed in one way, while the magnetic field due to the outer conductor is directed in the opposite way. Since the currents are equal and opposite, the magnetic fields will cancel each other out.

3. **Outside the Outer Conductor**: For a point outside the cable, we can apply Ampere's Law again. The total current enclosed by the Amperian loop in this region is zero because the equal and opposite currents cancel each other out.

### Step 5: Calculate the Magnetic Field
For a point outside the coaxial cable, the enclosed current \( I_{\text{in}} = I - I = 0 \). Thus, applying Ampere's Law:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \cdot 0 = 0 \]
This implies that the magnetic field \( B \) at any point outside the coaxial cable is zero.

### Conclusion
The magnetic field in the coaxial cable configuration, where the central and outer conductors carry equal currents in opposite directions, is zero outside the cable.

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