A very long, solid cylinder with radius R has positive...

A very long, solid cylinder with radius $R$ has positive charge uniformly distributed throughout it, with charge per unit volume $\rho$. (a) Derive the expression for the electric field inside the volume at a distance $r$ from the axis of the cylinder in terms of the charge density $\rho$. (b) What is the electric field at a point outside the volume in terms of the charge per unit length $\lambda$ in the cylinder? (c) Compare the answers to parts (a) and (b) for $r=R$. (d) Graph the electric-field magnitude as a function of $r$ from $r=0$ to $r=3 R$.

A very long, solid cylinder with radius $R$ has positive charge uniformly distributed throughout it, with charge per unit volume $\rho$.
(a) Derive the expression for the electric field inside the volume at a distance $r$ from the axis of the cylinder in terms of the charge density $\rho$.
(b) What is the electric field at a point outside the volume in terms of the charge per unit length $\lambda$ in the cylinder? (c) Compare the answers to parts (a) and (b) for $r=R$. (d) Graph the electric-field magnitude as a function of $r$ from $r=0$ to $r=3 R$.

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux through a closed surface to the charge enclosed by that surface. It is especially useful in situations with high symmetry, such as a uniformly charged cylinder, because it allows one to choose an appropriate Gaussian surface that simplifies the calculation by exploiting the symmetry of the charge distribution.

Cylindrical Symmetry

Cylindrical symmetry occurs when a system’s properties are invariant under rotations about a central axis and translations along that axis. For a long charged cylinder, this symmetry implies that the electric field depends only on the radial distance from the axis and points radially outward, greatly simplifying the analysis by reducing a potentially complex three-dimensional problem to a one-dimensional one.

Uniform Charge Density

Uniform charge density means that charge is distributed evenly throughout the volume of the cylinder, which allows the calculation of the enclosed charge within any Gaussian surface to be straightforwardly determined by volume integration. This key concept is essential for applying Gauss's Law, as it ensures the charge enclosed is directly proportional to the volume of the chosen Gaussian surface.

Boundary Conditions

Boundary conditions refer to the requirements that the electric field must satisfy at interfaces or surfaces, particularly where two different regions meet. In the context of a charged cylinder, matching the electric-field expressions at the surface (where the inside and outside fields converge) ensures that there are no discontinuities in the field unless a surface charge is present, providing consistency in the physical description.

Graphical Analysis of Electric Field

Graphical analysis involves plotting the magnitude of the electric field as a function of distance from the axis to visually interpret how the field changes both inside and outside the charged cylinder. This analysis not only helps in understanding the behavior of the field in different regimes, such as the linear increase inside the cylinder and the inverse distance dependency outside, but also reinforces the underlying principles of symmetry and Gauss's Law.

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