Coaxial Cable. A solid Figure E28.39 conductor with radius a...

Coaxial Cable. A solid Figure E28.39 conductor with radius $a$ is supported by insulating disks on the axis of a conducting tube with inner radius $b$ and outer radius $c$ (Fig. $\mathbf{E} 28.39$ ). The central conductor and tube carry equal currents $I$ in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (a) at points outside the central, solid conductor but inside the tube $(a<r<b)$ and $(b)$ at points outside the tube $(r>c) .$

Coaxial Cable. A solid Figure E28.39 conductor with radius $a$ is supported by insulating disks on the axis of a conducting tube with inner radius $b$ and outer radius $c$ (Fig. $\mathbf{E} 28.39$ ). The central conductor and tube carry equal currents $I$ in opposite directions. The currents are distributed
uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (a) at points outside the central, solid conductor but inside the tube $(a<r<b)$ and $(b)$ at points outside the tube $(r>c) .$

Key Concepts

Coaxial Cable Geometry

The coaxial cable setup, which consists of a central conductor and an outer cylindrical shell, takes advantage of cylindrical symmetry to simplify magnetic field calculations. When the conductors carry equal but opposite currents, the superposition of their magnetic fields leads to cancellation effects in certain regions, particularly outside the outer conductor. This geometric configuration is essential for understanding field distributions and the effectiveness of shielding in coaxial cables.

Magnetic Field Inside and Outside Conductors

In conductors with a uniformly distributed current, the magnetic field inside increases linearly with distance from the center because the enclosed current grows with the area. Conversely, for points outside the conductor, the entire current is enclosed, resulting in a magnetic field that typically decreases with increasing distance. This distinction is central to analyzing regions defined by different radial limits.

Uniform Current Distribution

When current is uniformly distributed over a conductor’s cross-sectional area, the current density remains constant across that area. This uniformity means that the amount of current enclosed by an Amperian loop is directly proportional to the area of the loop relative to the total cross-sectional area, leading to magnetic field expressions that vary in proportion to the radial distance inside the conductor.

Ampère’s Law

Ampère’s law relates the circulation of the magnetic field around a closed loop to the net current enclosed by that loop. This principle, derived from Maxwell’s equations, is especially useful for highly symmetric systems like cylindrical geometries, where it allows the magnetic field to be calculated in different regions by appropriately selecting an Amperian loop.

Transcript

Please give Ace some feedback