Problem 18 - Arihant AIEEE Physics

Question

Two negative charges of unit magnitude and a positive charge \(q\) are placed along a straight line. The charge \(q\) is placed between negative charges as such the system of charges is in equilibrium. This system is in : (a) stable equilibrium for the displacement of charge \(q\) in the normal direction of line joining the negative charges (b) unstable equilibrium for the displacement of charge \(q\) in the normal direction of line joining the negative charges (c) stable equilibrium for the displacement of charge \(q\) in the direction of line joining the negative charges (d) neutral equilibrium for the displacement of charge \(q\) along the line joining the negative charges

Step-by-Step Solution

Answer

(a) Stable equilibrium for normal displacements.

Step 1: Understanding Equilibrium

When a system of charges is in equilibrium, the net force acting on each charge is zero. In this problem, charge \( q \) is between two like charges (both negative and unit in magnitude) and is also in equilibrium. This indicates that the force exerted by each of the negative charges on \( q \) is equal in magnitude and opposite in direction, canceling each other out.

Step 2: Analyzing Forces in the Normal Direction

If \( q \) is displaced slightly in the direction perpendicular to the line joining the two negative charges, these forces no longer act directly along the line. Due to symmetry, the component forces in the perpendicular direction will attempt to return \( q \) to its original position. Hence, the equilibrium in the perpendicular direction is stable.

Step 3: Analyzing Forces along the Line

If \( q \) is displaced slightly in either direction along the line joining the negative charges, one force becomes stronger while the other becomes weaker due to proximity. This imbalance in forces will pull \( q \) further away from its original position, making the equilibrium unstable along the line.

Step 4: Final Conclusion

The system is stable for movements perpendicular to the line and unstable for movements along the line, meaning it's stable in one dimension and unstable in another. There is no scenario provided that fits this dual behavior exactly.

Key Concepts

Electric Forces

Electric forces are central to understanding interactions between charged particles. When charges are present, they exert forces on each other following Coulomb's Law. This law states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This force can either be attractive or repulsive. Two like charges (e.g., both positive or both negative) repel each other, while opposite charges attract.

In the given exercise, we have two negative charges and a positive charge placed along a line. The positive charge experiences repulsive forces from both negative charges. For the system to be in equilibrium, these forces must be equal in magnitude but opposite in direction, canceling each other out. This precise balance of electric forces keeps the system steady.

Stable vs Unstable Equilibrium

The concept of stable vs unstable equilibrium is crucial when analyzing systems of charges. **Stable equilibrium** refers to a state where, if a charge is slightly displaced, the forces acting upon it will try to bring it back to its original position. This state is like a ball at the bottom of a bowl; when displaced, gravity pulls it back down.

**Unstable equilibrium**, on the other hand, is when a slight displacement causes the charge to move further away from its initial position. Picture a ball balanced on top of a hill; a small nudge sends it rolling away. In our scenario, the charge positioned between two like charges finds itself in stable equilibrium when displaced perpendicular to their line of influence, as forces act to restore the charge to its equilibrium state. Conversely, it is in unstable equilibrium along the line joining the charges, as any slight movement disturbs the balance of forces, leading to further displacement.

Charge Interaction

Charge interactions are determined by the nature and proximity of the charges involved. In this exercise, two negative charges are positioned with a positive charge between them. The interaction between like charges (negative-negative) results in a repulsive force in this configuration.

Meanwhile, the positive charge interacts with both negative charges through electric forces, each pulling it in opposite directions. This tug-of-war establishes a peculiar situation where the net force on the positive charge is zero, achieving equilibrium. However, this equilibrium is contingent on the precise placement of the charges. Slight shifts either way alter their interaction, potentially destabilizing the system, indicating the delicate nature of electrostatic equilibrium.

Equilibrium Analysis

Equilibrium analysis helps understand the conditions necessary for a charge system to remain balanced. This involves examining the directions and magnitudes of forces acting on the charges. **Forces in Equilibrium**: When a system is in electrostatic equilibrium, the forces are balanced such that the net force is zero. It implies that the forces acting on each charge counterbalance one another precisely.

In our exercise, such analysis reveals that equilibrium in the line of action is unstable because minor shifts result in unequal forces due to changing distances. But when analyzing the system perpendicular to this line, stability is observed. Therefore, equilibrium analysis is crucial in predicting charge behavior under different conditions, leading to greater control and manipulation of electrostatic systems.