A very long insulating cylinder has radius R and carries...

A very long insulating cylinder has radius $R$ and carries positive charge distributed throughout its volume. The charge distribution has cylindrical symmetry but varies with perpendicular distance from the axis of the cylinder. The volume charge density is $\rho(r)=\alpha(1-r / R),$ where $\alpha$ is a constant with units $\mathrm{C} / \mathrm{m}^{3}$ and $r$ is the perpendicular distance from the center line of the cylinder. Derive an expression, in terms of $\alpha$ and $R,$ for $E(r),$ the electric field as a function of $r$. Do this for $r<R$ and also for $r>R$. Do your results agree for $r=R ?$

Key Concepts

Gaussian Surfaces

A Gaussian surface is a conceptual closed surface chosen to exploit the symmetry of a charge distribution. In the case of a long, charged cylinder with cylindrical symmetry, a coaxial cylindrical Gaussian surface simplifies the calculation of the electric field, as it ensures that the field has a constant magnitude over the surface, making the integration straightforward.

Volume Charge Density Integration

When dealing with continuous charge distributions, the total charge within a given region is obtained by integrating the volume charge density over that region. In problems where the charge density varies with position, such as varying linearly with the radial coordinate, integration is required to accurately calculate the charge enclosed by the Gaussian surface, which is then used in Gauss's Law.

Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is equal to the charge enclosed divided by the electric constant. This principle is fundamental in electrostatics as it relates the electric field to the distribution of charge, and it becomes particularly powerful when combined with symmetry arguments to simplify calculations.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical situation remains unchanged under rotations about the cylinder's axis and translations along the axis. This symmetry means that the electric field depends only on the radial distance from the axis, allowing the use of a cylindrical Gaussian surface to easily apply Gauss's Law and determine the field's behavior both inside and outside the charged cylinder.

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