A coaxial cable consists of a wire of radius a surrounded by...

A coaxial cable consists of a wire of radius $a$ surrounded by a thin metal cylindrical shell of radius $\bar{b}$. The wire has a uniform linear charge density $\lambda>0$ and the outer shell has a uniform linear charge density $-\lambda$. (a) Sketch the field lines for this cable. (b) Find expressions for the magnitude of the electric field in the regions $r \leq a, a<r<b$, and $\bar{b} \leq r$.

A coaxial cable consists of a wire of radius $a$ surrounded by a thin metal cylindrical shell of radius $\bar{b}$. The wire has a uniform linear charge density $\lambda>0$ and the outer shell has a uniform linear charge density $-\lambda$.
(a) Sketch the field lines for this cable. (b) Find expressions for the magnitude of the electric field in the regions $r \leq a, a<r<b$, and $\bar{b} \leq r$.

Key Concepts

Piecewise Analysis of Field Regions

Piecewise analysis involves breaking down the spatial domain into distinct regions where the behavior of the electric field can differ due to changes in the enclosed charge or material properties. In problems like those involving a coaxial cable, distinct regions—inside the charged conductor, in the gap between conductors, and outside the assembly—are analyzed separately using Gauss’s law to determine the appropriate expression for the electric field in each region.

Electric Field Lines

Electric field lines are a visual tool for representing the direction and relative magnitude of the electric field in space. They start on positive charges and end on negative charges, and their density in a given region depicts the strength of the field. In configurations with cylindrical symmetry like a coaxial cable, the field lines typically radiate outward or inward in the region between the conductors.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. In problems with high symmetry, it allows one to compute the electric field without directly integrating Coulomb’s law over the entire charge distribution, by choosing an appropriate Gaussian surface over which the electric field has a constant magnitude or predictable variation.

Cylindrical Symmetry

Cylindrical symmetry is a property of systems whose configurations are invariant under rotations about and translations along a central axis. In the context of a coaxial cable or similar problems, this symmetry allows the use of a cylindrical Gaussian surface, greatly simplifying the calculation of the electric field by reducing the problem to a dependence on the radial distance only.

Uniform Linear Charge Density

Uniform linear charge density indicates that the charge is evenly distributed along a line or a wire. This concept simplifies the analysis by ensuring that the contribution to the electric field from any segment of the charged object is constant along its length, allowing the use of symmetry and integration techniques that account only for the radial component of the electric field.

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