Applying Gauss' Law: Cylindrical Symmetry — Question 1

Question 1

The arbitrary Gaussian length cancels for an infinite line source

Use $\epsilon_0=8.854\times10^{-12}\,\mathrm{C^2/(N\,m^2)}$. An infinite straight line has uniform positive charge per length $\lambda=3.00$ nC/m. There are no other sources. Choose a coaxial Gaussian cylinder of radius r and length L.

1. Derive the field, explicitly treating side and end fluxes.

2. Calculate the field at $r=.200$ m. Compare the effects of doubling the Gaussian length and doubling the observation radius.

Question 1 - Solution

The arbitrary Gaussian length cancels for an infinite line source

(a) Method and calculation

$$\Phi_{\mathrm{ends}}=0,\quad\Phi_{\mathrm{side}}=E_r(2\pi rL),\quad Q_{\mathrm{enc}}=\lambda L.$$$$E_r=\frac{\lambda}{2\pi\epsilon_0r}.$$

(b) Interpretation and further calculation

$$E_r(.200)=\frac{3.00\times10^{-9}}{2\pi(8.854\times10^{-12})(.200)}\approx269.632\,\mathrm{N/C}.$$$$E_r(r;2L)=E_r(r;L),\quad E_r(2r)=\tfrac12E_r(r).$$L is an arbitrary surface-construction parameter, whereas r specifies the physical observation point.

Physical check

Radial direction and constant side-field magnitude follow from rotational and translational symmetry of the infinite source.