Problem 63 - Master Resource Book in JEE Main Physics

Question

Suppose, an imaginary cube is with a charge situated at the centre of it. The total electric flux passing through each of the faces of the cube will be (a) \(\frac{q}{6 \varepsilon_{0}}\) (b) \(\frac{q}{2 \varepsilon_{0}}\) (c) \(\frac{q}{12 \varepsilon_{0}}\) (c) None of these

Step-by-Step Solution

Answer

The flux through each face of the cube is \(\frac{q}{6 \varepsilon_{0}}\).

Step 1: Understanding the Problem

We are given a scenario with a charge located at the center of an imaginary cube. The problem asks for the total electric flux passing through each face of the cube.

Step 2: Understanding Electric Flux

The electric flux, represented as \(\Phi\), through a closed surface due to a point charge is given by Gauss's Law: \(\Phi = \frac{q}{\varepsilon_0}\), where \(q\) is the charge and \(\varepsilon_0\) is the vacuum permittivity.

Step 3: Applying Gauss's Law to the Cube

Since the charge is at the center of the cube, the electric flux is evenly distributed through all the cube's faces. The total flux through the entire surface of the cube is \(\frac{q}{\varepsilon_0}\).

Step 4: Finding the Flux Through One Face

A cube has 6 faces. Since the flux is uniformly distributed, the flux through each face is given by dividing the total flux by 6: \(\Phi_{face} = \frac{q}{6\varepsilon_0}\).

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism, connecting the electric flux emanating from a surface to the charge enclosed within that surface. It is mathematically expressed as \( \Phi = \frac{q}{\varepsilon_0} \), where \( \Phi \) represents the electric flux, \( q \) is the charge, and \( \varepsilon_0 \) is the vacuum permittivity. This law simplifies the calculation of electric fields when dealing with symmetrical charge distributions.When applying Gauss's Law, it is important to choose an appropriate closed surface, known as a Gaussian surface. The symmetry of the problem often guides this choice. For instance, when dealing with a point charge placed at the center of a cube, as in the exercise, the cube's surface serves as the Gaussian surface. The advantage of using Gauss's Law lies in its ability to bypass complex integration by exploiting symmetrical properties of electric fields.In simple terms, Gauss's Law allows us to calculate how much electric field (

• passes through an imaginary surface
• encloses a given charge distribution

without needing detailed knowledge of the field configuration within the surface.

Cubic Symmetry

Cubic symmetry refers to the situation where an object, such as a cube, and the charge distribution, such as a point charge at the center, have symmetrical properties. This symmetry can greatly influence the electric flux distribution across the surface of the cube. In the context of a cube with a point charge at its center, the symmetry of the cube plays a crucial role. Due to this symmetry:

• The electric field lines emanating from the charge are spread evenly across all directions.
• This results in an equal distribution of electric flux through each face of the cube.

Given this uniformity, the total flux from the charge is equally divided by the number of surfaces, in this case six (as the cube has six faces). Thus, the symmetry simplifies our calculations significantly by allowing us to assume that each face of the cube experiences an identical magnitude of electric flux. Understanding cubic symmetry is fundamental as it helps predict and calculate the behavior of electric fields in various practical applications, such as determining the force on charges positioned symmetrically around a central point.

Point Charge Distribution

A point charge distribution refers to a single point where all the charge is concentrated. In our problem scenario, a point charge is located precisely at the center of an imaginary cube. When dealing with point charges, one should:

• acknowledge that the electric field radiates outward symmetrically from the charge.
• realize that this is an idealized model, making the calculations easier under the assumption of a vacuum or infinite medium.

The placement of the point charge at the cube's center ensures that each face of the cube intercepts the same amount of electric field, due to the symmetrical radiation of electric lines. By understanding point charge distribution, you can better predict how electric fields will manifest in surrounding environments, aiding in various scientific and engineering applications. It simplifies the analysis by removing difficult variables, allowing more focus on symmetrical aspects of field distributions.