Problem 95 - Master Resource Book in JEE Main Physics

Question

A cube of side \(b\) has a charge \(q\) at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube. [NCERT] (a) \(\frac{4 q}{\sqrt{3} \pi \varepsilon_{0} b}\) (b) \(\frac{3 q}{\sqrt{2} \pi \varepsilon_{0} b}\) (c) \(\frac{3 q}{\sqrt{2} \pi \varepsilon_{0} b^{2}}\) (d) \(\frac{2 q}{\sqrt{3} \pi \varepsilon_{0} b}\)

Step-by-Step Solution

Answer

(a) \(\frac{4q}{\sqrt{3}\pi\varepsilon_{0}b\) and the electric field is zero.

Step 1: Understanding the Problem

We need to find the electric potential and electric field at the center of a cube with charges at each vertex. Given charge is represented as \(q\) and the side length of the cube is \(b\).

Step 2: Finding the Electric Potential at the Center

Each vertex of the cube has a charge \(q\). All charges contribute to the potential at the center, and potential \(V\) due to a point charge \(q\) at distance \(r\) is \(V = \frac{q}{4\pi\varepsilon_{0} r}\). The distance from the center of the cube to any vertex can be calculated using the Pythagorean theorem in 3D as \(r = \frac{b\sqrt{3}}{2}\). There are 8 vertices, so the total potential is \(V = 8 \times \frac{q}{4\pi\varepsilon_{0} \cdot \frac{b\sqrt{3}}{2}} = \frac{4q}{\sqrt{3}\pi\varepsilon_{0}b}\).

Step 3: Determining the Electric Field at the Center

The electric field \(\mathbf{E}\) due to a point charge \(q\) is \(\mathbf{E} = \frac{q}{4\pi\varepsilon_{0} r^2}\) and points away from the charge. However, since the cube is symmetric and the center is equidistant from all charges, the vector contributions of the electric field from each charge cancel out. Therefore, the net electric field at the center of the cube is zero.

Step 4: Concluding the Solution

The potential at the center due to charges at the vertices of the cube is \(\frac{4q}{\sqrt{3}\pi\varepsilon_{0}b}\), and the net electric field is zero. The correct choice based on potential is (a).

Key Concepts

Electric Field

The electric field is a concept in physics that helps us understand how electric charges influence the space around them. Imagine it as an invisible force field that surrounds any charged object. It can exert a force on other charges that enter its domain.
The magnitude and direction of this force field are described by a vector called the electric field vector, denoted as \(\mathbf{E}\).
- The formula for calculating the electric field due to a point charge \( q \) at a distance \( r \) is given by: \[\mathbf{E} = \frac{q}{4\pi\varepsilon_{0} r^2}\] - Here, \(\varepsilon_{0}\) is the permittivity of free space, essentially a constant that helps link the strength of the electric field to the amount of charge and distance.- The direction of the electric field vector points away from a positive charge and towards a negative charge
In the context of the cube with charges on its vertices, the symmetry ensures that all the electric field vectors cancel each other out at the center. This is why, despite each charge creating an electric field, the net electric field at the cube's center is zero.

Point Charge

A point charge is an idealized model of a charged object where the entire charge is considered to be concentrated at a single point in space. This simplification is incredibly useful in theories of electrostatics as it allows us to apply mathematical formulas more directly.
Understanding point charges helps in analyzing complex systems by breaking them down into simpler parts.
- For a point charge \( q \), the formula used to find its electric field at a distance \( r \) is: \[\mathbf{E} = \frac{q}{4\pi\varepsilon_{0} r^2}\]- Additionally, the electrical potential \( V \) it creates at a distance \( r \) is given by: \[V = \frac{q}{4\pi\varepsilon_{0} r}\]
In the scenario of the cube, each vertex behaves as a point charge. By calculating the effect of one and then summing them, we understand the entire system. Understanding the potential and electric field due to point charges allows us to solve the problem by considering these charges separately and utilizing symmetry.

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest. It focuses on how these charges create electric fields and interactions between them. At the foundation of electrostatics lie two essential laws: Coulomb's law for electric forces and Gauss's law for electric fields.
- Coulomb’s Law states that the electric force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. - Gauss’s Law helps us understand the distribution of electric charge over a surface and its corresponding electric field. A significant part of electrostatics involves calculating the electric potential and electric field created by a charge distribution. For the given exercise with a cube, electrostatic principles enable us to determine the influence of vertex charges on the cube’s center. It allows us to use symmetry and geometry to simplify otherwise complex calculations.

Charge Distribution

Charge distribution refers to the arrangement of electric charge on an object or within a system. In physics, understanding how charge is distributed helps us predict the electric field and potential created by that system.
This distribution can be uniform (same everywhere) or non-uniform (varying with position).
- For simple geometric configurations like spheres or lines, the symmetry of charge distribution can immensely simplify calculations. - In the context of the cube with charges at its vertices, each corner acts as a discrete point charge, creating a symmetrical charge distribution.
The symmetry in charge distribution is particularly helpful in this exercise because: - It simplifies calculations of potential at the center. - It ensures the electric field vectors from the corners cancel each other out. By recognizing the symmetry, we use the uniformity of distances from each vertex to the center of the cube to deduce the potential and net electric field. Thus, concepts of charge distribution combine with basic electrostatic principles to resolve complex charge configurations.