Problem 29 - Arihant AIEEE Physics

Question

Four equal positive charges each of magnitude \(q\) are placed at the respective vertices of a square of side length 1\. A point charge \(Q\) is placed at the centre of the square. Then : (a) \(Q\) must not be in equilibrium (b) \(Q\) must be in stable equilibrium (c) \(Q\) must be in neutral equilibrium (d) \(Q\) must be in unstable equilibrium

Step-by-Step Solution

Answer

Q is in neutral equilibrium.

Step 1: Understanding the system configuration

We have four equal positive charges placed at the corners of a square with side length 1 unit. A fifth charge, \(Q\), is located at the center of the square.

Step 2: Analyze forces on the central charge Q

The central charge \(Q\) experiences forces due to each of the charges located at the corners. Since all the corner charges are equal and symmetrically placed around the center, the forces due to charges at opposite corners will cancel each other out. Hence, the net force on \(Q\) is zero.

Step 3: Determine the nature of equilibrium

Since the net force on \(Q\) is zero, it implies that \(Q\) is in equilibrium. To determine if this equilibrium is stable, neutral, or unstable, consider a small displacement of \(Q\). If a small displacement results in a restoring force back to the original position, it is stable; otherwise, it is unstable. Here, any small displacement in the plane of the square would not result in a restoring force toward the center due to symmetry, thus indicating neutral equilibrium.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. Each charge exerts a force on the other, and the magnitude of this electrostatic force is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:\[F = k \frac{|q_1 q_2|}{r^2},\]where:

• \(F\) is the force between the charges,
• \(q_1\) and \(q_2\) are the magnitudes of the charges,
• \(r\) is the distance between the charges, and
• \(k\) is Coulomb's constant, due to the properties of the medium between the charges.

Coulomb's Law is fundamental in calculating forces in electrostatics. When dealing with multiple charges, like in the square configuration we have, superposition can be applied. This principle allows us to calculate the net force on a charge by summing up all the individual forces from other charges. Forces in opposite directions cancel out, and that's why the central charge \(Q\) in the exercise feels no net force initially.

Forces in a Symmetrical Configuration

Symmetry plays a crucial role in determining the resultant forces acting on a charge. In our exercise, four equal positive charges are placed at the vertices of a square. This arrangement creates a symmetrical setup around a central charge \(Q\) located at the center of the square.
The symmetrical disposition means that forces from charges at diagonally opposite corners of the square cancel each other out. Therefore, the attractive or repulsive forces exerted on the central charge by pairs of opposite charges neutralize the effect of each other.
It's essential to remember that even though each pair of opposite charges creates a force, the symmetry ensures that these forces balance out completely. Thus, in this particular symmetric configuration, a central charge experiences zero net force, as all the individual forces effectively nullify each other.

Types of Equilibrium in Physics

The concept of equilibrium is fundamental in physics, particularly concerning forces. Equilibrium refers to a state where the net force acting on an object is zero. There are three types of equilibrium: stable, neutral, and unstable.

• **Stable Equilibrium:** If a slight displacement of the object causes forces that return it to its original position. For example, a marble at the bottom of a bowl experiences this; upon displacement, gravity acts to restore it to the center.
• **Neutral Equilibrium:** Here, a minor displacement doesn't lead to a net restoring force. The object remains in its new position if moved. In our exercise, due to the symmetrical configuration of charges, when \(Q\) is slightly displaced within the plane of the square, no force compels it back to the center.
• **Unstable Equilibrium:** In this state, any small displacement results in forces that push the object further away. An example would be a pencil balanced on its tip.

Understanding these types helps predict how objects behave under various conditions of force. In our scenario, the charge \(Q\) is in neutral equilibrium because any slight displacement in the setup does not result in a restoring force to move it back to the center.