Question

A coaxial line carries current $i$ of uniform density in conductor of radius a (into the plane) and a uniform current $i$ through the outer conductor of inner radius b and outer radius c. (in a direction opposite to that of the inner current). The magnetic field at a distance $r$ from the axis, for $r > c$ is:

• A. $\frac{\mu_0 i}{2\pi r}$
• B. $\frac{\mu_0 i}{\pi r}$
• C. $\frac{\mu_0 i}{4\pi r}$
• D. 0

Answer 1

Answer: (D)

0

Answer 2

To find the magnetic field at a distance $r$ from the axis for $r > c$, we use Ampere's Law.

Ampere's Law states:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$$

1. Choose an Amperian Loop: For a coaxial cable, due to its cylindrical symmetry, the magnetic field lines are concentric circles. We choose a circular Amperian loop of radius $r$ concentric with the axis, such that $r > c$.

2. Calculate the enclosed current ($I_{enclosed}$):

   • The inner conductor carries a current $i$ into the plane. Let's consider current into the plane as positive for now. So, $I_{inner} = +i$.

   • The outer conductor carries a current $i$ in a direction opposite to that of the inner current. This means the current in the outer conductor is out of the plane. So, $I_{outer} = -i$.

   The total current enclosed by the Amperian loop (which is outside both conductors, $r > c$) is the algebraic sum of the currents in the inner and outer conductors:

   $$I_{enclosed} = I_{inner} + I_{outer} = i + (-i) = 0$$

3. Apply Ampere's Law:

   The magnetic field $\vec{B}$ is tangential to the circular Amperian loop and has a constant magnitude $B$ along the loop. The line integral $\oint \vec{B} \cdot d\vec{l}$ simplifies to $B \cdot (2\pi r)$.

   Substituting the enclosed current into Ampere's Law:

   $$B (2\pi r) = \mu_0 (0)$$ $$B (2\pi r) = 0$$

   Since $2\pi r$ cannot be zero (as $r > c$ and $c > 0$), the magnetic field $B$ must be zero.

   $$B = 0$$

This result signifies that a coaxial cable effectively confines the magnetic field within its outer conductor, preventing it from extending outside.