Figure 23-40 shows a section of a long, thin-walled metal...

Figure $23-40$ shows a section of a long, thin-walled metal tube of radius $R=3.00 \mathrm{cm},$ with a charge per unit length of $\lambda=2.00 \times 10^{-8} \mathrm{C} / \mathrm{m} .$ What is the magnitude $E$ of the electric field at radial distance (a) $r=R / 2.00$ and (b) $r=2.00 R ?(\mathrm{c})$ Graph $E$ versus $r$ for the range $r=0$ to 2.00$R .$

Figure $23-40$ shows a section of a
long, thin-walled metal tube of radius
$R=3.00 \mathrm{cm},$ with a charge per unit
length of $\lambda=2.00 \times 10^{-8} \mathrm{C} / \mathrm{m} .$ What
is the magnitude $E$ of the electric field
at radial distance (a) $r=R / 2.00$ and
(b) $r=2.00 R ?(\mathrm{c})$ Graph $E$ versus $r$
for the range $r=0$ to 2.00$R .$

Key Concepts

Field Calculation for a Line Charge

In regions outside a charged conductor that can be approximated by a line charge, the electric field can be calculated by treating the charge as if it were concentrated along a line. The field at a distance from the line charge falls off inversely with the radial distance, a result derived from Gauss's Law using a cylindrical Gaussian surface that encloses the entire charge per unit length.

Electric Field Inside a Conductor

For conductors in electrostatic equilibrium, the electric field within the conducting material is zero. In the case of a charged thin-walled metal tube, the free electrons move such that any internal electric field is canceled. This results in no net electric field in the region enclosed by the conductor, a fact that is essential when applying Gauss's Law to points inside the charged surface.

Graphing Electric Field Discontinuities

Graphing the electric field as a function of radial distance involves understanding the behavior of the field in different regions—inside the conductor where it is zero, and outside where it follows an inverse relationship with distance. The transition at the surface of the conductor may show a discontinuity in the slope of the graph, reflecting the sudden change in the enclosed charge as one moves from inside to outside the charged tube.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. This principle is fundamental in calculating electric fields, especially in problems exhibiting symmetry. It states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space, allowing a simplification of field calculations when the charge distribution has a high degree of symmetry.

Cylindrical Symmetry

Cylindrical symmetry is a feature of charge distributions that extend infinitely (or are long relative to their dimensions) in one direction, resulting in electric fields that depend only on the distance from the axis. This symmetry permits the choice of a cylindrical Gaussian surface to simplify the integration of the electric field over the surface, reducing complex 3D problems to simpler radial calculations.

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