Saurabh Koranglekar
Last Activity: 5 Years ago
To understand the situation with the two charged spheres suspended in a liquid, we need to apply some principles from electrostatics and fluid mechanics. The problem states that the spheres are identical and maintain a consistent angle of 30° with each other when submerged in a liquid. Let's break this down step by step to find the dielectric constant of the liquid.
Analyzing the Forces Acting on the Spheres
Each sphere experiences several forces: the gravitational force acting downward, the tension in the string acting along the string, and the electrostatic force due to the charges on the spheres. When suspended, these forces must be in equilibrium.
Forces at Play
- Weight of the sphere (W): This can be calculated using the formula W = m * g, where m is the mass of the sphere and g is the acceleration due to gravity.
- Tension in the string (T): This force acts along the length of the string and has both vertical and horizontal components.
- Electrostatic Force (F): The force between the two charged spheres can be described by Coulomb's law: F = k * (|q1 * q2|) / r², where k is Coulomb's constant, q1 and q2 are the charges on the spheres, and r is the distance between the centers of the spheres.
Equilibrium of Forces
For the system to be in equilibrium, the vertical components of the tension must balance the weight of the spheres, and the horizontal components of the tension must balance the electrostatic force.
Weight of the Sphere Calculation
The density of the material of the spheres is given as 1.6 g/cm³. The mass can be calculated by multiplying the volume of the sphere by its density. The volume (V) of a sphere is given by the formula V = (4/3)πr³. However, since we do not have the radius, we can express the weight in terms of density:
W = density * volume * g = 1.6 g/cm³ * V * g.
Understanding the Angle and Tension Components
Let's denote the angle between the strings as θ (which is 30°). The vertical component of the tension can be expressed as:
T_vertical = T * cos(θ)
And the horizontal component as:
T_horizontal = T * sin(θ)
Deriving the Dielectric Constant
Given that the spheres maintain the same angle in the liquid, we can conclude that the buoyant force acting on each sphere is equal to the weight of the sphere. The buoyant force (B) in a liquid is given by:
B = density_liquid * V * g = 0.8 g/cm³ * V * g.
Since the weight of the sphere in the liquid must equal the buoyant force plus the vertical tension, we can write:
1.6 g/cm³ * V * g = 0.8 g/cm³ * V * g + T * cos(30°).
Now, we can also relate the electrostatic force to the tension in the horizontal direction:
T * sin(30°) = k * (|q1 * q2|) / r².
Given that the spheres are in a liquid with a dielectric constant (κ), the electrostatic force in the liquid is modified as:
F = (1/κ) * k * (|q1 * q2|) / r².
To find κ, we must establish a relationship between these forces, the density of the liquid, and the electrostatic properties of the spheres. The dielectric constant can be derived by rearranging the equations above, factoring in the buoyancy and tension relationships.
Final Calculation of the Dielectric Constant
Using the equilibrium equations and solving for κ, we find that the dielectric constant of the liquid is:
κ = (T * sin(30°)) / (T * sin(30°) - (1.6 g/cm³ - 0.8 g/cm³) * V * g).
When you input the specific values for the variables, you can solve for the dielectric constant. This approach not only highlights the equilibrium conditions of forces but also integrates the concepts of buoyancy and electrostatics effectively.
