Question

A single positive point charge, \(q,\) is at one corner of a cube with sides of length \(L\), as shown in the figure. The net electric flux through the three net electric flux through the three adjacent sides is zero. The net electric flux through each of the other three sides is a) \(q / 3 \epsilon_{0}\). b) \(q / 6 \epsilon_{0}\). c) \(q / 24 \epsilon_{0}\). d) \(q / 8 \epsilon_{0}\).

Short answer

Answer: The net electric flux through three non-adjacent faces of the cube is \(\frac{q}{16\epsilon_{0}}\).

Step-by-step solution

Understand Gauss's law

To solve this problem, we need to recall Gauss's law: \(\Phi = \frac{Q_{enc}}{\epsilon_0}\), where \(\Phi\) is the electric flux, \(Q_{enc}\) is the charge enclosed by the surface, and \(\epsilon_0\) is the permittivity of free space.

Analyze the symmetry

In this case, the charge \(q\) is placed at the corner of a cube. We can imagine a larger cube that has 8 times the volume of the original cube and the charge at its center. The larger cube would be symmetric, and each of its 8 smaller cubes will have the same electric flux through their respective portions of the faces shared with other cubes.

Applying Gauss's law to the larger cube

Since the larger cube is symmetric and has a charge \(q\) at its center, we can directly apply Gauss's law to it: \(\Phi_{large} = \frac{q}{\epsilon_0}\). The total electric flux through the larger cube's surface is equal to the charge at its center divided by \(\epsilon_0\).

Divide the total electric flux by 8

Now since the larger cube can be divided into 8 equal smaller cubes, the electric flux through the surfaces of identical faces of these smaller cubes will be equal. Thus, the net electric flux through three non-adjacent faces of the original cube will be the electric flux associated with one of these 8 smaller cubes through half its faces (three non-adjacent faces): \(\Phi_{small} = \frac{\Phi_{large}}{2\times 8} = \frac{\frac{q}{\epsilon_0}}{16}\).

Find the correct answer

The net electric flux through each of the other three sides is \(\Phi_{small} = \frac{q}{16\epsilon_{0}}\). Comparing this answer with the given options, we find that none of the options match our result. Thus, the correct answer is not listed among the given options.

Key concepts

Gauss's Law

Understanding Gauss's Law is essential for analyzing situations involving electric fields and charges. In its simplest form, Gauss's Law states that the net electric flux, \( \Phi \), through a closed surface is directly proportional to the enclosed electric charge, \( Q_{enc} \), within that surface. Mathematically, it is expressed as:
\[\Phi = \frac{Q_{enc}}{\epsilon_0}\]
Here, \(\epsilon_0\) represents the permittivity of free space, a constant that characterizes the strength of the electrical interaction in a vacuum.
To conceptualize this, imagine wrapping an imaginary bubble, known as a Gaussian surface, around a charge. The flux through this bubble tells us how much field 'flows' through the surface, akin to counting the number of field lines crossing it. For symmetric arrangements, like spheres, cylinders, or planes, Gauss's Law simplifies calculations significantly and is more convenient than Coulomb's Law.

Net Electric Flux

Net electric flux is a measure of the total field passing through a surface. It's important to acknowledge that flux can be both entering and exiting a surface. Should the field uniformly enter one side and exit the opposite, the net flux would be zero, suggesting no 'excess' electric field in one direction over the other.
In scenarios with a point charge near a cube, as stated in the exercise, calculating the net electric flux can be more complex due to asymmetric field distribution. By considering a larger, symmetric situation and breaking it into smaller identical sections, as we do when applying Gauss's Law to the large cube and then dividing its flux, we can determine the flux through a part of an asymmetric figure.
Consequently, the net electric flux is particularly revealing. It either shows the amount of the electric field 'leaking' out (if the charge is inside the surface) or the effective field impacting the surface (if the charge is outside or partially enclosed). This idea is pivotal to understanding how fields interact with surfaces and is widely used in deriving principles like capacitance or in grounding electromagnetic theory.

Permittivity of Free Space

The permittivity of free space, denoted as \(\epsilon_0\), is a fundamental constant that describes how much electric field is 'permitted' to pass through the vacuum of free space. It essentially sets the scale for electric force and flux in the universe and has a value of approximately \(8.854 \times 10^{-12}\) farads per meter (F/m).
When dealing with electric flux and Gauss's Law, \(\epsilon_0\) is a critical factor appearing in the denominator of the law's equation. Its role is to adjust the magnitude of the charge to yield the correct units and actual electric flux value through a surface. A useful analogy is to think of \(\epsilon_0\) as a 'filter'—it determines how strongly the vacuum resists the electric field lines trying to pass through it. Lower permittivity would mean less resistance to electric flux and vice versa. Its understanding is not only essential for electric flux calculations but also for defining electric potential and understanding dielectric materials where permittivity differs from the vacuum value.