A steady current \(I\) flows in one direction in the solid inner conductor, radius \(a\), of a coaxial cable, and in the opposite direction in the outer conductor whose inner radius is \(b\) and outer radius \(c\). Find the magnetic field at various distances from the axis.

When learning about the magnetic field in a coaxial cable, understanding Ampere's Law is fundamental. It is a crucial tool that tells us how electric current creates magnetic fields. The law is elegantly described by the equation \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed} \), where \(\vec{B}\) is the magnetic field, \(d\vec{l}\) is an infinitesimal element of a closed loop, and \(I_{enclosed}\) is the current enclosed by the loop. In simpler terms, Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through that loop.

In the context of the coaxial cable problem, Ampere's Law is applied to determine the magnetic field in different regions relative to the cable's cross-section. By using a circular path coaxial with the cable, Ampere's Law helps us calculate the magnetic field based on the current distribution. This means, for different radii within the cable, we take loop paths of corresponding radii, which help us uncover how the magnetic field varies across different distances from the central axis of the cable.

In practice, we often simplify the use of Ampere's Law by selecting symmetrical paths where the magnetic field strength is constant, making it easier to solve the integrals involved. This is exactly what was done in the textbook solution for the coaxial cable, simplifying a potentially complex integration to a straightforward multiplication.

Calculating magnetic fields is a fundamental part of electromagnetism and is key to understanding how forces act on charges in motion. In our coaxial cable example, the magnetic field calculation depends on the region being considered.

The step-by-step solution from the textbook demonstrates that Ampere's Law is employed thrice, once for each distinct region within the coaxial cable setup. The mathematical execution involves integrating the magnetic field along a closed path—circular loops were chosen due to the symmetry of the problem. This approach reflects a powerful method in physics: exploiting symmetry to reduce complexity.

In regions where current is enclosed, the magnetic field varies depending on the distance from the axis, which is related to the amount of current enclosed by the chosen loop. Calculations involve determining this enclosed current and applying it to Ampere's Law to obtain the field. Understanding this process is critical as it provides insight into how fields emanate and interact with materials. Notably, the calculations reveal that inside the inner conductor, the field increases linearly with radius, is inversely proportional to the radius between the conductors, and outside the outer conductor—owing to the opposing currents nullifying each other—the field drops to zero.

Grasping the concept of current distribution is imperative for understanding the magnetic fields inside and around conductors. In essence, the way electric current is distributed across a conductor directly influences the resultant magnetic field. This is crucial when dealing with the coaxial cable problem, where the current flows through concentric conductors in opposite directions.

Inside the Inner Conductor

The inner conductor in our coaxial cable problem carries a uniform current density throughout its cross-section. When calculating magnetic fields within this region, it's essential to consider only the portion of current that lies inside the circular path defined by the radius \(r\). This portion increases quadratically with the radius, explaining why the enclosed current—and hence the resulting magnetic field—also increases linearly with \(r\) within the inner conductor.

Between the Conductors

In the annular space between the inner and outer conductors, the entire current from the inner conductor is within any circular path we choose (unless \(r\) is bigger than the outer radius). This gives us a unified current that sets up a magnetic field decreasing with distance, portraying an inverse relationship with \(r\).

Outside the Outer Conductor

Finally, outside the outer conductor, an intriguing scenario unfolds. The currents in the two conductors flow in opposite directions, leading to equal and opposite contributions to the net current enclosed by paths in this region. Since these currents cancel each other, the net magnetic field is zero, reflecting a fundamental aspect of current distribution influence on magnetic fields.