LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Coulomb's Law
The Setup in Air
Imagine two identical charged spheres hanging from a common point. Because they carry the same charge, they repel each other. Eventually, they settle into a state of equilibrium where the strings make an angle with each other.
Let's analyze the forces acting on one of these spheres in the air. There are three primary forces at play:
1. The downward gravitational force, or weight:
2. The horizontal electrostatic repulsion:
3. The tension in the string:
For the sphere to be in equilibrium, the horizontal and vertical components of these forces must balance perfectly. Resolving the tension into its components, we get:
By dividing these two equations, we eliminate the tension and obtain a beautiful, simple relationship for the angle:
The Plot Twist
Enter the Liquid
Now, the entire setup is submerged in a liquid with a density . The problem presents a fascinating constraint: the angle remains exactly the same! How is this possible? Let's look at what changes when the system is in the liquid.
First, the electrostatic force is weakened because the liquid acts as a dielectric medium. The new electric force is:
where is the dielectric constant of the liquid.
Second, the liquid exerts an upward buoyant force on the spheres, effectively reducing their weight. According to Archimedes' principle, the buoyant force is equal to the weight of the displaced liquid:
The effective weight of the sphere is its actual weight minus this buoyant force:
Since the mass of the sphere is , we can rewrite the effective weight as:
The Grand Equating
In the liquid, the new equilibrium condition gives us a new equation for :
Because the problem states that the angle remains unchanged, we can equate our two expressions for :
Notice how the original electric force and the weight cancel out perfectly from both sides! This leaves us with a pure relationship between the dielectric constant and the densities:
Rearranging to solve for :
Finally, we substitute the given values for the densities ( and ):
The dielectric constant of the liquid is exactly 2.
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