Question

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

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

Answer 1

The Correct Option is C

By applying Ampère's law:

\[ \oint \vec{B} \cdot d\vec{\ell} = \mu_0 i_{\text{enc}} = 0 \]

Since the net enclosed current outside the cable is zero (equal and opposite currents in the central and outer conductors), the magnetic field outside the cable is zero.

Answer 2

Step 1: Understanding the setup
A coaxial cable has a central conductor and an outer cylindrical conductor. Both carry equal currents in opposite directions. The magnetic field at any point depends on the distribution of current and the distance from the axis of the cable.

Step 2: Apply Ampere’s law
According to Ampere’s law: \[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enclosed}}. \] For different regions:

• Inside the inner conductor: The field increases linearly with radius \(r\) as \(B = \frac{\mu_0 I r}{2\pi R_1^2}\) (assuming uniform current).
• Between the two conductors: The enclosed current is \(I\) (from the inner conductor), so \(B = \frac{\mu_0 I}{2\pi r}\).
• Inside the outer conductor: The net enclosed current decreases linearly until it becomes zero at the outer surface.

Step 3: Outside the outer conductor
For any point outside the cable (radius greater than the outer conductor), the total enclosed current is: \[ I_{\text{net}} = I - I = 0. \] Hence, by Ampere’s law, the magnetic field \(B = 0\) outside the cable.

Step 4: Physical interpretation
The magnetic fields generated by the inner and outer conductors cancel completely in the region outside because their currents are equal in magnitude and opposite in direction. Therefore, there is no net magnetic field outside the coaxial cable.

Final answer

outside the cable