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

Two long, charged, thin-walled, concentric cylindrical shells have radii of $3.0$ and $6.0 \mathrm{~cm}$. The charge per unit length is $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $+5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ 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 in a system with multiple charge distributions, the total electric field at a point is the vector sum of the electric fields produced by each individual charge distribution. In the context of concentric cylindrical shells, the electric field at a point is determined by summing the contributions from each shell based on the amount of charge enclosed by the Gaussian surface at that particular radial distance.

Electric Field of a Charged Cylinder

For an infinitely long, charged cylindrical shell or line charge, the electric field is derived by applying Gauss's Law to a cylindrical Gaussian surface. The resulting electric field at a distance from the cylinder is inversely proportional to the radial distance from the axis if the charge is uniformly distributed, which is key for analyzing fields both inside and outside the charged cylinders.

Gauss's Law

This fundamental law relates the net electric flux through a closed surface to the charge enclosed by that surface. In problems involving cylindrical charge distributions, Gauss's Law is particularly useful because it allows one to calculate the electric field by choosing a Gaussian surface that matches the symmetry of the system, thereby simplifying the integration to a product of the electric field and the surface area.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical properties, such as the electric field, depend only on the radial distance from the axis of the cylinder and not on the angular or axial coordinates. This symmetry permits the use of a coaxial cylindrical Gaussian surface, which helps in determining that the field is uniformly radial and simplifies the calculations.

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