Tension In A Uniformly Charged String Ring Due To Charge Present On It

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Introduction

In the realm of electrostatics, the study of charged particles and their interactions is a fundamental concept. One of the intriguing problems in this field is the calculation of tension in a uniformly charged string ring due to electrostatic repulsion of charge present on it. This problem has been a subject of interest for physicists and mathematicians alike, and in this article, we will delve into the details of this problem and provide a step-by-step solution.

Problem Statement

Consider a uniformly charged ring with a total charge Q distributed evenly along its circumference. The ring is assumed to be thin and flexible, and we want to calculate the tension in the ring due to the electrostatic repulsion of the charge present on it.

Mathematical Formulation

To solve this problem, we need to consider the electrostatic potential energy of the ring. Let's denote the charge density on the ring as λ, and the radius of the ring as R. The electrostatic potential energy of the ring can be calculated using the formula:

U = (1/2) ∫(λ/R) dθ

where θ is the angle subtended by the element of charge at the center of the ring.

Simplifying the Integral

To simplify the integral, we can use the fact that the charge density is uniform along the circumference of the ring. This means that the charge density can be expressed as:

λ = Q / (2πR)

Substituting this expression into the integral, we get:

U = (1/2) ∫(Q / (2πR^2)) dθ

Evaluating the Integral

To evaluate the integral, we can use the fact that the integral of 1/cos(x) is cosec(x). In this case, we have:

∫(1/cos(x)) dx = cosec(x)

Using this result, we can rewrite the integral as:

U = (1/2) ∫(Q / (2πR^2)) dθ = (1/2) (Q / (2πR^2)) ∫dθ = (1/2) (Q / (2πR^2)) θ

Calculating the Tension

The tension in the ring is given by the force per unit length. To calculate the tension, we need to take the derivative of the potential energy with respect to the radius of the ring. Using the chain rule, we get:

T = dU/dR = (1/2) (Q / (2πR^2)) dθ/dR = (1/2) (Q / (2πR^2)) (2πR) = Q / (2R)

Conclusion

In this article, we have calculated the tension in a uniformly charged string ring due to electrostatic repulsion of charge present on it. We have used the electrostatic potential energy of the ring to derive an expression for the tension, and have evaluated the integral using the fact that the charge density is uniform along the circumference of the ring. The result shows that the tension in the ring is proportional to the total charge Q and inversely proportional to the radius R of the ring.

Additional Information

If you have to solve this problem before and have encountered difficulties with integrating cosec(x), you are not alone. The integral of cosec(x) is a challenging one, and requires careful manipulation of the expression. However, with the correct approach and a bit of patience, you can arrive at the correct solution.

References

  • [1] Griffiths, D. J. (2017). Introduction to Electrodynamics. Pearson Education.
  • [2] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.

Further Reading

If you are interested in learning more about electrostatics and the calculation of tension in charged systems, we recommend the following resources:

  • [1] "Electrostatics" by MIT OpenCourseWare
  • [2] "Classical Electrodynamics" by Stanford University
  • [3] "Electromagnetism" by University of Colorado Boulder

Q: What is the tension in a uniformly charged string ring due to electrostatic repulsion of charge present on it?

A: The tension in a uniformly charged string ring due to electrostatic repulsion of charge present on it is given by the formula:

T = Q / (2R)

where Q is the total charge on the ring and R is the radius of the ring.

Q: What is the electrostatic potential energy of the ring?

A: The electrostatic potential energy of the ring is given by the formula:

U = (1/2) ∫(λ/R) dθ

where λ is the charge density on the ring and θ is the angle subtended by the element of charge at the center of the ring.

Q: How do you calculate the tension in the ring?

A: To calculate the tension in the ring, you need to take the derivative of the potential energy with respect to the radius of the ring. Using the chain rule, you get:

T = dU/dR = (1/2) (Q / (2πR^2)) dθ/dR = (1/2) (Q / (2πR^2)) (2πR) = Q / (2R)

Q: What is the significance of the radius of the ring in calculating the tension?

A: The radius of the ring plays a crucial role in calculating the tension. As the radius of the ring increases, the tension decreases, and vice versa. This is because the electrostatic repulsion between the charges on the ring decreases as the distance between them increases.

Q: Can you provide an example of how to calculate the tension in a uniformly charged string ring?

A: Let's consider a ring with a total charge Q = 10 μC and a radius R = 0.1 m. Using the formula for tension, we get:

T = Q / (2R) = 10 μC / (2 x 0.1 m) = 50 N

Q: What are some common applications of the concept of tension in a uniformly charged string ring?

A: The concept of tension in a uniformly charged string ring has several applications in physics and engineering, including:

  • Calculating the force between charged particles
  • Studying the behavior of charged systems
  • Designing electrical circuits and devices
  • Understanding the properties of materials under electrostatic stress

Q: Are there any limitations or assumptions in the calculation of tension in a uniformly charged string ring?

A: Yes, there are several limitations and assumptions in the calculation of tension in a uniformly charged string ring, including:

  • The ring is assumed to be thin and flexible
  • The charge density is uniform along the circumference of the ring
  • The electrostatic repulsion between the charges on the ring is assumed to be the only force acting on the ring
  • The ring is assumed to be in a vacuum or a medium with negligible electrostatic properties

Q: Can you provide any additional resources or references for further reading on this topic?

A: Yes, here are some additional resources and references for further reading on this topic:

  • [1] Griffiths, D. J. (2017). Introduction to Electrodynamics. Pearson Education.
  • [2] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
  • [3] "Electrostatics" by MIT OpenCourseWare
  • [4] "Classical Electrodynamics" by Stanford University
  • [5] "Electromagnetism" by University of Colorado Boulder

We hope this FAQ article has provided a helpful summary of the key concepts and calculations involved in determining the tension in a uniformly charged string ring due to electrostatic repulsion of charge present on it. If you have any further questions or comments, please feel free to contact us.