6.3 Applying gauss’s law  (Page 6/13)

where the direction information is included by using the unit radial vector.

Charge distribution with cylindrical symmetry

A charge distribution has cylindrical symmetry    if the charge density depends only upon the distance r from the axis of a cylinder and must not vary along the axis or with direction about the axis. In other words, if your system varies if you rotate it around the axis, or shift it along the axis, you do not have cylindrical symmetry.

[link] shows four situations in which charges are distributed in a cylinder. A uniform charge density ${\rho }_0.$ in an infinite straight wire has a cylindrical symmetry, and so does an infinitely long cylinder with constant charge density ${\rho }_0.$ An infinitely long cylinder that has different charge densities along its length, such as a charge density ${\rho }_1$ for $z>0$ and ${\rho }_2\ne {\rho }_1$ for $z<0$ , does not have a usable cylindrical symmetry for this course. Neither does a cylinder in which charge density varies with the direction, such as a charge density ${\rho }_1$ for $0\le \theta <\pi$ and ${\rho }_2\ne {\rho }_1$ for $\pi \le \theta <2\pi$ . A system with concentric cylindrical shells, each with uniform charge densities, albeit different in different shells, as in [link] (d), does have cylindrical symmetry if they are infinitely long. The infinite length requirement is due to the charge density changing along the axis of a finite cylinder. In real systems, we don’t have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in, then the approximation of an infinite cylinder becomes useful.

Consequences of symmetry

In all cylindrically symmetrical cases, the electric field ${\overrightarrow{\text{E}}}_P$ at any point P must also display cylindrical symmetry.

Cylindrical symmetry: ${\overrightarrow{\text{E}}}_P=E_P(r)\widehat{\text{r}}$ ,

where r is the distance from the axis and $\widehat{\text{r}}$ is a unit vector directed perpendicularly away from the axis ( [link] ).

Gaussian surface and flux calculation

To make use of the direction and functional dependence of the electric field, we choose a closed Gaussian surface in the shape of a cylinder with the same axis as the axis of the charge distribution. The flux through this surface of radius s and height L is easy to compute if we divide our task into two parts: (a) a flux through the flat ends and (b) a flux through the curved surface ( [link] ).