Problem 170 - A Complete Resource Book in Physics for JEE Main

Question

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of \(30^{\circ}\) with each other. When suspended in a liquid of density \(8 \mathrm{~g} \mathrm{~cm}^{-3}\), the angle remains the same. If density of the material of the sphere is \(16 \mathrm{~g} \mathrm{~cm}^{-3}\), the dielectric constant of the liquid is (A) 4 (B) 3 (C) 2 (D) 1

Step-by-Step Solution

Answer

The dielectric constant of the liquid is \(K = 2\). Hence, the correct answer is (C) 2.

Step 1: Calculate the buoyant force acting on the spheres

To determine the buoyant force acting on the suspended spheres, we need to use the Archimedes' principle, which states that the buoyant force is equal to the weight of the liquid displaced by the spheres. Buoyant force, F_B = weight of the liquid displaced = volume of liquid displaced × density of liquid × acceleration due to gravity Let the volume of each sphere be V, then the volume of liquid displaced is also V. The density of the liquid is given as 8 g/cm³, and we can assume the acceleration due to gravity to be approximately 9.81 m/s² or 981 cm/s². Hence, the buoyant force acting on each sphere is: F_B = V × 8 g/cm³ × 981 cm/s²

Step 2: Calculate the tension in the strings

The spheres are in equilibrium under the action of three forces: the tension in the string (T), the repulsive electric force between the spheres (F_R), and the buoyant force (F_B). Since the angle between the strings remains the same when the spheres are suspended in the liquid, we can equalize the forces acting on the spheres in both cases - air and liquid. In the vertical direction, the tension should balance the weight and the buoyant force: Tsin(30°) = weight of the sphere (mg) - F_B Now, the mass (m) can be expressed as the product of the volume (V) and the density of the sphere's material (ρ_sphere), i.e., m = ρ_sphere × V. The density of the sphere's material is given as 16 g/cm³, and g = 981 cm/s². Therefore, Tsin(30°) = V × 16 g/cm³ × 981 cm/s² - V × 8 g/cm³ × 981 cm/s²

Step 3: Calculate the repulsive force between the spheres

In the horizontal direction, the tension should balance the repulsive electric force between the spheres. Tcos(30°) = F_R Now, we are going to substitute the value of Tsin(30°) from Step 2 into this equation: (V × 16 g/cm³ × 981 cm/s² - V × 8 g/cm³ × 981 cm/s²) / cos(30°) = F_R Since we are interested in the repulsive force between the spheres in the liquid, we can replace F_R with F_R_liquid.

Step 4: Find the dielectric constant of the liquid

The repulsive force between the charges in a medium is inversely proportional to the dielectric constant of that medium. We can use the following relationship to find the dielectric constant (K) of the liquid: F_R_air / F_R_liquid = K From Step 3, we have the equation for F_R_liquid. Therefore, we can rearrange this equation and find the dielectric constant: K = F_R_air / F_R_liquid = (V × 16 g/cm³ × 981 cm/s² - V × 8 g/cm³ × 981 cm/s²) / [(V × 8 g/cm³ × 981 cm/s²) × cos(30°)] After simplifying the expression, we get the dielectric constant K: K = 2 Hence, the correct answer is (C) 2.