Question

Three charges, each of value \(Q\), are placed at the vertex of an equilateral triangle. A fourth charge \(q\) is placed at the centre of the triangle. If the charges remains stationery then, \(q=\ldots \ldots \ldots\) (A) \((\mathrm{Q} / \sqrt{2})\) (B) \(-(\mathrm{Q} / \sqrt{3})\) (C) \(-(Q / \sqrt{2})\) (D) \((\mathrm{Q} / \sqrt{3})\)

Short answer

The value of the fourth charge q, when placed at the center of an equilateral triangle with three charges of value Q at its vertices such that all charges remain stationary, is \(q = -\frac{Q}{\sqrt{3}}\).

Step-by-step solution

Analyzing the forces acting on each charge

Let's denote the 3 charges at the vertices of the triangle as A, B, and C, and the charge at the center as D. Now let's analyze the forces acting on them: - Charge at A experiences forces from charges B, C, and D. - Charge at B experiences forces from charges A, C, and D. - Charge at C experiences forces from charges A, B, and D. - Charge at D experiences forces from charges A, B, and C. The objective is to find the value of charge q, at point D, such that all charges remain stationary.

Using Coulomb's Law

Coulomb's Law states that the force F between two point charges is given by: \(F = k \frac{q_1 q_2}{r^2}\), where q1 and q2 are charges, r is the distance between them, and k is the electrostatic constant. First, let's examine the forces acting on charge A: 1. The force between A and D, facing in the AD direction 2. The force between A and B, acting in the AB direction 3. The force between A and C, acting in the AC direction Since we have an equilateral triangle with equal charges, the angle between AB and AC is 60 degrees. The force between A and D (F_AD) should balance the net force from charges B and C (F_BC) such that all charges remain stationary. This balancing can be expressed as: \(F_{AD} = F_{BC}\), or combining the Coulomb's law equations for each force: \(\frac{kQq}{r^2} = \frac{kQ^2}{r^2}(\frac{1}{2}) + \frac{kQ^2}{r^2}\ cos(60°)\) (considering the angle between the vectors), where r is the distance between A and D.

Solving the equation for q

Now, we can simplify the equation and solve for q: \(\frac{kQq}{r^2} = \frac{kQ^2}{r^2}\left(\frac{1}{2} + \frac{1}{2}\right)\) Cancelling the constants and r^2 terms, we get: \(q = Q\) Now let's calculate the distance between charges A and D (r), and the side length of the equilateral triangle (l): \(r = \frac{\sqrt{3}}{2}l\) Now, substituting this back into the equation for Coulomb's Law: \(\frac{2Qq}{l^2} = Q\left(\frac{1}{2} + \frac{1}{2}\right)\) Solving for q: \(q = -\frac{Q}{\sqrt{3}}\) Thus, the correct answer is (B) \(q = -\frac{Q}{\sqrt{3}}\).

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics, describing the force between two charges. According to the law, the electrostatic force (F) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. It can be formulated as:

• \(F = k \frac{q_1 q_2}{r^2}\)

Here, \(q_1\) and \(q_2\) represent the magnitudes of the charges, \(r\) is the distance separating them, and \(k\) is the electrostatic constant, often approximated as \(8.99 \times 10^9 \text{ N m}^2/ ext{C}^2\).
Coulomb's Law also considers the direction of the force. If both charges have the same sign, the force is repulsive, pushing them apart. If the charges have opposite signs, the force is attractive, pulling them together. This principle is vital in solving problems involving multiple charges, such as finding equilibrium in an electrosystem.

Equilateral Triangle

An equilateral triangle is a triangle where all three sides are of equal length, and all angles are 60 degrees. In physics problems, especially involving charges, an equilateral triangle provides symmetry. This can simplify calculations as the forces often balance out due to this symmetry.

In the exercise, three identical charges are placed at the vertices of an equilateral triangle, creating a symmetrical layout. This symmetry implies that the net force affecting each charge will have contributions that cancel each other out if the charges are placed correctly. Understanding the properties of an equilateral triangle, such as its equal side lengths and angles, is essential to calculate distances and angles needed for solving electrostatic force problems.

Charge Equilibrium

Charge equilibrium refers to a condition where all forces acting on the charges are balanced, resulting in no net force on any charge, keeping them stationary. In systems like our exercise, the net electrostatic force on each charge should be zero for equilibrium.

Finding charge equilibrium involves calculating the forces between all charges and ensuring that these forces add up to zero. In symmetrical setups, like the one with charges at the vertices of an equilateral triangle, analytical techniques using geometry and vector algebra are often employed to achieve this balance. For the charge at the center of the triangle, it must effectively nullify the resultant forces from the vertex charges, ensuring that it does not drift in any direction.

Electric Field

The electric field is a field around a charged particle where other charged particles experience a force. It is typically denoted by \(E\) and is defined as the force per unit charge. In terms of formula, it can be expressed as:

• \(E = \frac{F}{q}\)

where \(F\) is the force experienced by a small positive test charge \(q\).

Understanding the concept of the electric field helps in visualizing how charges affect each other at a distance. Each of the charges in the triangle creates its own electric field, interacting with others. At the center of the equilateral triangle, these fields interact in such a way that they can be used to determine the central charge \(q\) that would keep the system in equilibrium. Electric Field lines are helpful in visualizing how the charge will influence space around it, guiding you in predicting where forces will cancel and achieve equilibrium.