Question

A \(30.0-\mathrm{cm}\) -long uniformly charged rod is sealed in a container. The total electric flux leaving the container is \(1.46 \cdot 10^{6} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}\). Determine the linear charge distribution on the rod.

Short answer

Answer: The linear charge distribution of the rod is \(4.3 \times 10^{-5} \mathrm{~C/m}\).

Step-by-step solution

Write down the given information

We are given the following information: - Length of the rod (L): \(30.0 \mathrm{~cm}\) or \(0.3 \mathrm{~m}\) - Total electric flux (\(\Phi_E\)): \(1.46 \times 10^6 \mathrm{~N~m^2/C}\) - The linear charge distribution of the rod: \(\lambda\) (unknown)

Express Gauss's law

According to Gauss's Law: \(\Phi_E = \frac{Q_{enclosed}}{\epsilon_0}\) Where \(\Phi_E\) is the electric flux, \(Q_{enclosed}\) is the total charge enclosed, and \(\epsilon_0\) is the vacuum permittivity constant (\(\epsilon_0 \approx 8.85 \times 10^{-12} \mathrm{~C^2/N~m^2}\))

Find the total charge enclosed

We can calculate the total charge enclosed in the container by rearranging Gauss's law: \(Q_{enclosed} = \Phi_E \times \epsilon_0\) Now plug in the given values: \(Q_{enclosed} = (1.46 \times 10^6 \mathrm{~N~m^2/C}) \times (8.85 \times 10^{-12} \mathrm{~C^2/N~m^2})\) \(Q_{enclosed} = 1.29 \times 10^{-5} \mathrm{~C}\)

Determine the linear charge distribution on the rod

The linear charge distribution, \(\lambda\), is equal to the total charge enclosed, \(Q_{enclosed}\), divided by the length of the rod, \(L\): \(\lambda = \frac{Q_{enclosed}}{L}\) Now plug in the values of \(Q_{enclosed}\) and \(L\): \(\lambda = \frac{1.29 \times 10^{-5} \mathrm{~C}}{0.3 \mathrm{~m}}\) \(\lambda = 4.3 \times 10^{-5} \mathrm{~C/m}\) So, the linear charge distribution of the rod is \(4.3 \times 10^{-5} \mathrm{~C/m}\).

Key concepts

Linear Charge Distribution

Understanding the concept of linear charge distribution is fundamental when dealing with electrostatics. This refers to the amount of electric charge per unit length along a line or, as in our exercise, along a conductive rod. When we say that a rod has a uniform linear charge distribution, it means that the charge is spread evenly along its entire length.

In practical terms, this uniform distribution simplifies calculations because it allows us to treat the charge as being concentrated at evenly distributed points along the rod. This is important for solving many problems in electrostatics, including using Gauss's Law to find electric fields around charged objects.

The unit of linear charge distribution, often denoted as \(\lambda\), is coulombs per meter (C/m). It reflects how much charge is found in a very small piece of the rod we call an 'element' of length. Once we know the total charge and the length of the rod, finding \(\lambda\) is a straightforward division, enhancing our understanding of the electric properties of the rod.

Gauss's Law

Gauss's Law is a cornerstone of electromagnetism and plays a critical role in our understanding of electric fields. In essence, it provides a connection between the electric flux flowing out of a closed surface and the charge enclosed within that surface. Mathematically expressed, Gauss's Law is written as \(\Phi_E = \frac{Q_{enclosed}}{\epsilon_0}\), where \(\Phi_E\) represents the electric flux, \(Q_{enclosed}\) is the total charge within the enclosed surface, and \(\epsilon_0\) is the vacuum permittivity.

This law suggests that regardless of the shape or size of the surface, the total electric flux through a closed surface is proportional to the enclosed charge. This concept becomes particularly useful with symmetrical charge distributions because it allows us to calculate electric fields and charge distributions even when the geometry appears complicated.

In layman's terms, Gauss's Law can be seen as an elegant statement about field lines: for every charge inside the surface, there must be a corresponding number of electric field lines passing through the surface, balancing out the enclosed charge.

Vacuum Permittivity

Vacuum permittivity, denoted by the symbol \(\epsilon_0\), is a fundamental constant that characterizes the ability of the vacuum to permit electric field lines. Technically, it's the measure of resistance encountered when forming an electric field in a vacuum. It's a relatively small number, approximately equal to \(8.85 \times 10^{-12} \mathrm{~C^2/N~m^2}\), which shows the weak electrical force per unit charge in free space.

This constant plays a pivotal role in Gauss's Law, as it helps balance the equation by accounting for the effects of the medium in which the charges reside—in this case, a vacuum. To put it simply, \(\epsilon_0\) is a proportionality factor that determines the strength of the electric field produced by a given electric charge in a vacuum. In most problems, including our exercise, \(\epsilon_0\) is a given value that's critical when calculating the enclosed charge or the electric flux in situations where the medium is air or a vacuum.