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Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid .
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Two identical balls each having density σ are suspended from a common...
Given:
- Two identical balls with density σ are suspended from a common point by two insulating strings of equal length.
- Both balls have equal mass and charge.
- In equilibrium, each string makes an angle θ with the vertical.
- Both balls are immersed in a liquid with density ρ.
- The angle θ does not change after immersing the balls in the liquid.
To find:
- The dielectric constant of the liquid (ε).
Solution:
1. Analysis of forces:
- In equilibrium, the weight of each ball is balanced by the tension in the string and the electrostatic force between the balls.
- The weight of each ball can be calculated as W = mg, where m is the mass of each ball.
- The tension in each string can be calculated as T = mg/cos(θ), where θ is the angle the strings make with the vertical.
- The electrostatic force between the balls can be calculated as Fe = kq^2/r^2, where k is the Coulomb's constant, q is the charge on each ball, and r is the separation between the balls.
2. Analysis of forces in the liquid:
- When the balls are immersed in the liquid, they experience buoyant force in addition to the forces mentioned above.
- The buoyant force on each ball can be calculated as Fb = ρVg, where V is the volume of each ball.
- The volume of each ball can be calculated as V = (4/3)πr^3, where r is the radius of each ball.
- The weight of each ball in the liquid can be calculated as Wl = (m - ρV)g.
3. Equilibrium condition:
- The angle θ does not change after immersing the balls in the liquid, which indicates that the net force in the vertical direction is still zero.
- The net force in the vertical direction can be calculated as:
Fnet = T - Wl - Fb = 0
T - (m - ρV)g - ρVg = 0
4. Calculation of the dielectric constant:
- From the equilibrium condition, we can solve for V:
T - (m - ρV)g - ρVg = 0
T - mg + ρVg - ρVg = 0
T - mg - ρVg = 0
ρV = (T - mg)/g
- The electrostatic force between the balls can be written in terms of the dielectric constant:
Fe = kεq^2/r^2
- Equating the electrostatic force to the tension in the string:
kεq^2/r^2 = T
- Substituting for T and ρV:
kεq^2/r^2 = (T - mg)/g
kεq^2/r^2 = ((mg/cos(θ)) - mg)/g
kεq^2/r^2 = m(1 - cos(θ))
- Simplifying the equation:
ε = (1 - cos(θ))r^2/kq^2
Therefore, the dielectric constant of the
- Two identical balls with density σ are suspended from a common point by two insulating strings of equal length.
- Both balls have equal mass and charge.
- In equilibrium, each string makes an angle θ with the vertical.
- Both balls are immersed in a liquid with density ρ.
- The angle θ does not change after immersing the balls in the liquid.
To find:
- The dielectric constant of the liquid (ε).
Solution:
1. Analysis of forces:
- In equilibrium, the weight of each ball is balanced by the tension in the string and the electrostatic force between the balls.
- The weight of each ball can be calculated as W = mg, where m is the mass of each ball.
- The tension in each string can be calculated as T = mg/cos(θ), where θ is the angle the strings make with the vertical.
- The electrostatic force between the balls can be calculated as Fe = kq^2/r^2, where k is the Coulomb's constant, q is the charge on each ball, and r is the separation between the balls.
2. Analysis of forces in the liquid:
- When the balls are immersed in the liquid, they experience buoyant force in addition to the forces mentioned above.
- The buoyant force on each ball can be calculated as Fb = ρVg, where V is the volume of each ball.
- The volume of each ball can be calculated as V = (4/3)πr^3, where r is the radius of each ball.
- The weight of each ball in the liquid can be calculated as Wl = (m - ρV)g.
3. Equilibrium condition:
- The angle θ does not change after immersing the balls in the liquid, which indicates that the net force in the vertical direction is still zero.
- The net force in the vertical direction can be calculated as:
Fnet = T - Wl - Fb = 0
T - (m - ρV)g - ρVg = 0
4. Calculation of the dielectric constant:
- From the equilibrium condition, we can solve for V:
T - (m - ρV)g - ρVg = 0
T - mg + ρVg - ρVg = 0
T - mg - ρVg = 0
ρV = (T - mg)/g
- The electrostatic force between the balls can be written in terms of the dielectric constant:
Fe = kεq^2/r^2
- Equating the electrostatic force to the tension in the string:
kεq^2/r^2 = T
- Substituting for T and ρV:
kεq^2/r^2 = (T - mg)/g
kεq^2/r^2 = ((mg/cos(θ)) - mg)/g
kεq^2/r^2 = m(1 - cos(θ))
- Simplifying the equation:
ε = (1 - cos(θ))r^2/kq^2
Therefore, the dielectric constant of the
Community Answer
Two identical balls each having density σ are suspended from a common...
Let Theta=Q and sigma = S TcosQ= mg TsinQ= coulomb force between the balls Make a free body diagram and find tangent of theta. Now take the second case in which balls are immersed in liquid. In this case we would take upthrust(=vSg) acting on the ball. Now TcosQ= mg + vSg TsinQ=(coulomb force between the balls)/K K is the dielectric constant of liquid. Again find tangent of theta. Since angle does not change so equate the two values of tangent of theta. Answer would come out K= P/(P-S)
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Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid .
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Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid ..
Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid ..
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Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid ., a detailed solution for Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . has been provided alongside types of Two identical balls each having density σ are suspended from a common point by two insulating strings of equal length. Both the balls have equal mass and charge. In equilibrium each string makes an angle θ with vertical. Now both the balls are immersed in liquid. But the angle does not change. the density of the liquid is ρ. find the dielectric constant of the liquid . theory, EduRev gives you an
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