ILW Two long, charged, thin-walled, concentric cylindrical...

ILW Two long, charged, thin-walled, concentric cylindrical shells have radii of $3.0$ and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer shell. What are the (a) magnitude $E$ and (b) direction (radially inward or outward) of the electric field at radial distance $r=4.0 \mathrm{~cm} ?$ What are (c) $E$ and (d) the direction at $r=8.0 \mathrm{~cm}$ ?

Key Concepts

Superposition Principle

The superposition principle states that the total electric field produced by multiple charge distributions is the vector sum of the electric fields created by each individual distribution. This principle is essential when dealing with systems that include more than one charged object, as it allows the independent calculation of the field contributions from each object and their subsequent algebraic summing.

Enclosed Charge

The concept of enclosed charge refers to the net amount of electric charge contained within a chosen Gaussian surface. When applying Gauss’s Law, only the charge inside the Gaussian surface contributes to the electric flux, which is crucial for determining the electric field in regions both inside and outside the charge distributions.

Gauss's Law

Gauss's Law states that the total electric flux through any closed surface is equal to the total charge enclosed within that surface divided by the permittivity of free space. This principle is fundamental in calculating electric fields in situations with high symmetry because it relates the field to the enclosed charge without requiring detailed integration over the charge distribution.

Cylindrical Symmetry

Cylindrical symmetry occurs when a system of charges or current distributions has symmetry around a central axis, meaning the physical properties do not change with rotations around that axis. This symmetry allows the selection of an appropriate Gaussian surface, typically a co-axial cylinder, that simplifies the calculation of the electric field by reducing complex integrals to simpler expressions.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in conjunction with Gauss’s Law to simplify the calculation of the electric field. In the context of cylindrical shells, choosing a cylindrical Gaussian surface that shares the same axis as the shells exploits the symmetry of the problem, making it possible to evaluate the electric flux and relate it directly to the enclosed charge.

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