Problem 55 - Master Resource Book in JEE Main Physics

Question

Three charges each of magnitude \(q\) are placed at the corners of an equilateral triangle. The electrostatic force on the charge placed at the centre is (a) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{q^{2}}{L^{2}}\) (b) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{3 q^{2}}{L^{2}}\) (c) \(\frac{1}{12 \pi \varepsilon_{0}} \frac{q^{2}}{L^{2}}\) (d) zero

Step-by-Step Solution

Answer

The net electrostatic force on the charge placed at the center is zero.

Step 1: Understanding the Problem

We have an equilateral triangle with charges of magnitude \(q\) at each corner. We need to determine the net electrostatic force on a charge placed at the center of the triangle due to these three charges.

Step 2: Conceptual Insight into Electrostatic Force

According to Coulomb's Law, the magnitude of the force between two point charges is given by \( F = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2} \). In our case, all charges are equal, and we need to consider how forces from each charge at the corners add up at the center.

Step 3: Symmetry Consideration

Because the triangle is equilateral and all charges are the same, the forces due to each charge will have the same magnitude, but their directions will be uniformly spread out. If we consider symmetry, any force from one charge will be equally counteracted by the resultant of the other two forces.

Step 4: Force Calculation and Resultant Force

Assuming the distances from the corners to the center are equal, due to symmetry, the vector sum of the forces from each charge on the central charge results in zero. This is because the triangle's symmetric nature means the forces are equally distributed and cancel each other out.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism that describes the electrostatic interaction between electrically charged particles. This law states that the force (\[ F \]) between two point charges is directly proportional to the product of the magnitudes of the charges (\[ q_1 \], \[ q_2 \]) and inversely proportional to the square of the distance (\[ r \]) between them:

• Formula: \( F = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2} \)
• Constant \( \varepsilon_0 \) represents the permittivity of free space.

This formula is essential for calculating the electrostatic force in various geometrical arrangements. In the context of an equilateral triangle, it helps determine how each charge at the corners influences a charge positioned at the triangle's center.
This law also highlights that the force is attractive if charges have opposite signs and repulsive if they have the same sign. By understanding this law, students can grasp how forces between charges behave in different configurations.

Equilateral Triangle

An equilateral triangle is a geometric shape where all three sides and angles are equal. This symmetry is crucial in understanding the distribution of forces in problems involving electrostatic forces. When point charges are placed at each corner of an equilateral triangle:

• Each charge exerts an equal magnitude of force on any charge placed at the center.
• The distances from each corner to the center are identical, maintaining uniformity.

The equilateral nature means that the central charge experiences forces from each corner charge, which are symmetrical in magnitude and distribution. Such symmetry plays a vital role in analyzing and simplifying the net forces acting on a charge, reducing what could be a complex vector addition problem into a simpler one due to cancels of forces caused by symmetry.
In electrostatic force problems, like the one described, knowing the properties of the equilateral triangle allows students to use symmetry as a shortcut to derive results, rather than doing detailed calculations for each force component.

Vector Addition

Vector addition is a process used to determine the resultant force when multiple forces are acting on an object. Each force is represented as a vector, having both magnitude and direction. In the context of charges at the corners of an equilateral triangle:

• The forces due to each charge on a central charge must be summed vectorially.
• Direction and angle of each force are essential for accurate vector addition.

To add vectors, you typically arrange them head-to-tail or use components when they are at different angles. With symmetrical setups, like our equilateral triangle problem, sometimes the resultant force calculation can simplify drastically. Here, symmetry ensures that forces can cancel out, meaning the vector addition process leads to zero net force.
In conclusion, mastering vector addition and understanding symmetry allows students to handle complex force problems effectively, reducing intimidating scenarios to basic, discernible outcomes.

Symmetry in Physics

Symmetry in physics refers to a situation where a system remains unchanged under certain transformations, such as shifts, rotations, or reflections. In the realm of electrostatics, symmetry makes complex calculations more manageable. For the given problem with charges at the corners of an equilateral triangle:

• Symmetry ensures each charge generates a force with equal magnitude on the central charge.
• The triangle's symmetry means these forces are distributed uniformly.

Using symmetry, it's easy to predict that the net force at the center is zero. The geometry of an equilateral triangle causes the forces to cancel out completely because they are equal in magnitude and symmetrically opposed.
Understanding symmetry helps recognize that often the sum of forces in such configurations does not require extensive computation. This concept not only simplifies problem solving but also underlines fundamental principles that govern physical laws across multiple areas in science.