Question

The inward and outward electric flux for a closed surface in units of \(\mathrm{Nm}^{2} / \mathrm{C}\) are respectively \(8 \times 10^{3}\) and \(4 \times 10^{3}\). Then the total charge inside the surface is \(\ldots \ldots \ldots \ldots . . \mathrm{c}\). (A) \(\left[\left(-4 \times 10^{3}\right) / \epsilon_{0}\right]\) (B) \(-4 \times 10^{3}\) (C) \(4 \times 10^{3}\) (D) \(-4 \times 10^{3} \mathrm{E}_{0}\)

Short answer

The total charge inside the surface is approximately \(\left[-4 \times 10^{3}/\epsilon_{0}\right]\), which is closest to option (A). However, the exact value of the total charge enclosed is \(-3.54 \times 10^{-8} \ \mathrm{C}\).

Step-by-step solution

Given information

We are given the inward and outward electric fluxes for a closed surface. The inward electric flux is \(8 \times 10^3 \ \mathrm{Nm}^2/\mathrm{C}\), and the outward electric flux is \(4 \times 10^3 \ \mathrm{Nm}^2/\mathrm{C}\).

Applying Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface divided by the permittivity of free space (\(\epsilon_0\)). Mathematically, it can be written as: \[\Phi_E = \frac{Q_{enclosed}}{\epsilon_0}\] Where \(\Phi_E\) is the total electric flux through the closed surface, \(Q_{enclosed}\) is the total charge enclosed by the surface, and \(\epsilon_0\) is the permittivity of free space (\(\epsilon_0 \approx 8.85 \times 10^{-12} \ \mathrm{C}^2/\mathrm{Nm}^2\)).

Find the net electric flux

We need to find the net electric flux through the closed surface. Since the inward flux is negative and the outward flux is positive, we subtract the inward flux from the outward flux: \[\Phi_{net} = \Phi_{outward} - \Phi_{inward}\] \[\Phi_{net} = 4\times 10^3 - 8\times 10^3 = - 4\times 10^3 \ \mathrm{Nm}^2/\mathrm{C}\]

Calculate the total charge enclosed

Now, we can use Gauss's Law to find the total charge enclosed by the surface: \[Q_{enclosed} = \Phi_{net} \cdot \epsilon_0\] \[Q_{enclosed} = (-4 \times 10^3 \ \mathrm{Nm}^2/\mathrm{C}) \cdot (8.85 \times 10^{-12} \ \mathrm{C}^2/\mathrm{Nm}^2)\] \[Q_{enclosed} = -3.54 \times 10^{-8} \ \mathrm{C}\] We can see that the total charge enclosed by the surface is closest to option (A): (A) \(\left[-4 \times 10^{3}/\epsilon_{0}\right]\) However, the \(Q_{enclosed}\approx -3.54 \times 10^{-8}\), so the correct answer is closest to, but not exactly equal to option (A).

Key concepts

Electric Flux

Electric flux is an essential concept in electromagnetism, particularly when working with Gauss's Law. It measures the number of electric field lines passing through a given area. Electric flux can be visualized as how much of the electric field "flows" through a surface.
The formula for electric flux through a surface is given by:

• \(\Phi_E = \mathbf{E} \cdot \mathbf{A} = E \cdot A \cdot \cos\theta\)

Here, \(\mathbf{E}\) is the electric field, \(\mathbf{A}\) is the area vector perpendicular to the surface, \(E\) is the magnitude of the electric field, \(A\) is the area of the surface, and \(\theta\) is the angle between \(\mathbf{E}\) and \(\mathbf{A}\).
If the field lines are perpendicular to the surface, flux is maximized, whereas if they run parallel, flux is zero. This idea helps in calculating how much electric charge is enclosed within a specific region, as part of Gauss's Law. In our exercise, the net electric flux is calculated by considering both inward and outward fluxes passing through a closed surface.

Closed Surface

A closed surface serves an important role in physics, particularly in understanding electrostatic principles through Gauss's Law. Imagine a surface that completely encloses a volume, like a sphere or a cube, with no openings. This is what we refer to as a closed surface in the context of electric flux.
The significance of a closed surface comes from its ability to illustrate the flow of electric flux around charges:

• Within a closed surface, electric field lines can enter and exit, illustrating inward and outward fluxes.
• Gauss's Law states that the net electric flux through a closed surface is directly related to the total charge contained within that surface.

In the provided exercise, a closed surface is used to determine the net electric flux by accounting for all field lines crossing the boundary. Through this method, we can calculate the total charge enclosed by understanding both the positive (outward) and negative (inward) contributions to electric flux.

Permittivity of Free Space

Permittivity of free space, often symbolized as \(\epsilon_0\), is a fundamental constant that plays a key role in electromagnetism. It serves as a measure of how much electric field (or force) is "permitted" through the vacuum free of electric charges. The value of permittivity of free space is approximately \(8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{Nm}^2\).
Permittivity is crucial because:

• It appears in the denominator of Coulomb's Law, influencing the force between two charges in vacuum.
• In Gauss's Law, it relates the total charge inside a closed surface to the resulting electric flux.

In the context of the exercise, we use \(\epsilon_0\) to calculate the total charge within a closed surface from the net electric flux. The relationship is given by the formula \(Q_{enclosed} = \Phi_{net} \cdot \epsilon_0\), demonstrating how permittivity directly connects electric flux and charge in a closed system. Understanding this constant helps decipher the relationship between electric charges and fields.