Question

A cubical region of side \(a\) has its centre at the origin. It encloses three fixed point charges, \(-q\) at \((0,-a / 4,0),+3 q\) at \((0,0,0)\) and \(-q\) at \((0,+a / 4,0)\). Choose the correct option(s). (A) The net electric flux crossing the plane \(x=+a / 2\) is equal to the net electric flux crossing the plane \(x=-a / 2\). (B) The net electric flux crossing the plane \(y=+a / 2\) is more than the net electric flux crossing the plane \(y=-a / 2\) (C) The net electric flux crossing the entire region is \(\frac{q}{\varepsilon_{0}}\). (D) The net electric flux crossing the plane \(z=+a / 2\) is equal to the net electric flux crossing the plane \(x=+a / 2\).

Short answer

The net electric flux through planes \(x=+a/2\) and \(x=-a/2\); \(y=+a/2\) and \(y=-a/2\); and \(z=+a/2\) and \(z=-a/2\) is equal due to symmetry (Options A and D). The net electric flux through the entire region is \(\frac{q}{\varepsilon_{0}}\) (Option C). Option B is incorrect because Gauss's law does not depend on the distribution of charge within the surface.

Step-by-step solution

Identify the Symmetry and Apply Gauss's Law

Gauss's law states that the net electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space \( \varepsilon_0 \). Because of symmetry and the arrangement of charges, the flux through each face of the cube should be the same due to equal distribution of enclosed charges.

Calculate Net Charge Inside the Cube

Sum the charges inside the cube to find the net charge. The total charge is \( -q + 3q -q = q \).

Calculate Flux through Planes Parallel to x-Axis

Using symmetry, we can say that the planes \(x=+a/2\) and \(x=-a/2\) will have the same net electric flux because the total charge is symmetrically distributed along the x-axis.

Calculate Flux through Planes Parallel to y-Axis

Since the charges are not symmetrically distributed along the y-axis, the net electric flux through planes \(y=+a/2\) and \(y=-a/2\) cannot be determined by symmetry alone and would ideally require electric field calculations. However, Gauss's law only depends on the enclosed charge, not its distribution, so these planes will also have the same net electric flux.

Calculate Net Electric Flux through the Entire Region

The electric flux through the entire cube is given by \( \Phi = \frac{q}{\varepsilon_{0}} \), from the net charge inside the cube. This applies to the entire region, not specific planes.

Calculate Flux through Planes Parallel to z-Axis

Using symmetry again, the net electric flux through the planes \(z=+a/2\) and \(z=-a/2\) will be equal because the cube and charge distribution are symmetric along the z-axis.

Key concepts

Electric Flux

Electric flux is a fundamental concept used in electromagnetism to describe how an electric field interacts with an area in space. Imagine electric field lines passing through a surface; the electric flux quantifies the number of these lines. It's directly proportional to the strength of the electric field and the area it passes through, and it also depends on the angle between the field lines and the surface. Mathematically, electric flux (\( \Phi_E \)) is defined by the equation \( \Phi_E = E \cdot A \cdot \cos(\theta) \), where \( E \) is the electric field strength, \( A \) is the area of the surface, and \( \theta \) is the angle between the field lines and the perpendicular to the surface.

For a uniform electric field and a flat surface, this concept simplifies as the product of the field and the area's perpendicular component. Let's apply this to our cubical region problem: since we are dealing with a cube, the surface area relevant for electric flux calculations is that of one side of the cube, \( a^2 \). Given that symmetry plays a crucial role in this specific problem, and the cube's orientation with respect to the charges, the calculation becomes focused on understanding how symmetry dictates the same flux through opposite planes.

Charge Distribution

Charge distribution refers to how charge is spatially arranged in a system. It’s critical in determining the electric field within and around that system. There are different types of charge distributions, such as point charges, continuous charge distributions, and symmetrical charge distributions, each having unique implications for the resulting electric field.

In our cube example, there are three point charges with specific locations within the cube. These locations impact the symmetry of the charge distribution. Understanding that the charge distribution within this cube is not uniform—as two negative \(-q\) charges are positioned at the midpoints along one axis with a positive \(+3q\) charge at the origin—helps us in further calculations. However, according to Gauss's law, the net through any closed surface only depends on the total charge enclosed within the surface. This principle guides us in determining that, despite the non-uniform distribution, the net electric flux through any side of the cube will correspond solely to the sum of the charges inside.

Permittivity of Free Space

The permittivity of free space, denoted as \( \varepsilon_0 \), is a constant used in the calculation of electric fields and forces in a vacuum. It represents how much resistance is encountered when forming an electric field in the vacuum. The value of \( \varepsilon_0 \) is approximately \( 8.85 \times 10^{-12} \frac{C^2}{N \cdot m^2} \). It plays a significant role in Gauss's law, as it is the constant of proportionality that relates electric flux to the net charge enclosed by a surface.

In the context of our cube problem, \( \varepsilon_0 \) is key to calculating the net electric flux through the cubical region for the entire charge distribution. Once we have the net charge (by summing individual charges), we can determine the net electric flux (\( \Phi \)) through the entire cube by \( \Phi = \frac{q}{\varepsilon_0} \), independent of the specific geometry of the distribution or the orientation of the cube itself, as dictated by Gauss's law.