A Charge Q Is To Be Divided On Two Small Conducting Spheres. What Should Be The Value Of Charges On The Spheres So That When Placed At A Certain Distance Apart, The Repulsive Force Between Them Is Maximum?
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Introduction
When two small conducting spheres are placed at a certain distance apart, they can either attract or repel each other, depending on the charges they carry. In this scenario, we are tasked with dividing a charge q onto two small conducting spheres in such a way that the repulsive force between them is maximized. This problem is a classic example of electrostatics and requires a deep understanding of the principles governing the behavior of charged particles.
Theoretical Background
To approach this problem, we need to recall the fundamental principles of electrostatics. When two charged objects are placed near each other, they experience a force due to the interaction between their electric fields. The magnitude and direction of this force depend on the charges carried by the objects, as well as the distance between them.
The repulsive force between two charged objects can be calculated using Coulomb's Law, which states that the force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, this can be expressed as:
F = k * (q1 * q2) / r^2
where F is the force between the charges, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between them.
Maximizing Repulsive Force
To maximize the repulsive force between the two spheres, we need to consider the following factors:
- Charge Distribution: The charge q should be divided between the two spheres in such a way that the resulting charges are as large as possible.
- Distance: The distance between the spheres should be minimized to maximize the repulsive force.
Charge Distribution
Let's assume that the charge q is divided between the two spheres in the ratio x:1-x, where x is a variable between 0 and 1. This means that one sphere carries a charge of qx, while the other sphere carries a charge of q(1-x).
Using Coulomb's Law, we can calculate the repulsive force between the two spheres as a function of x:
F(x) = k * (qx * q(1-x)) / r^2
To maximize this force, we need to find the value of x that maximizes the expression.
Maximizing the Repulsive Force
To maximize the repulsive force, we can take the derivative of F(x) with respect to x and set it equal to zero:
dF/dx = k * (q^2 * (1-2x)) / r^2 = 0
Solving for x, we get:
x = 1/2
This means that the charge q should be divided equally between the two spheres, with each sphere carrying a charge of q/2.
Conclusion
In conclusion, to maximize the repulsive force between two small conducting spheres, the charge q should be divided equally between the two spheres, with each sphere carrying a charge of q/2. This is achieved by minimizing the distance between the spheres and dividing the charge in such a way that the resulting charges are as large as possible.
References
- Coulomb, C. A. (1785). Theorie des affections electriques et du magnetisme. Paris: Imprimerie Royale.
- Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press.
Further Reading
- Electromagnetism: Principles and Applications by I. D. Abrahams
- Classical Electrodynamics by J. D. Jackson
Related Topics
- Electrostatics
- Coulomb's Law
- Electric Fields
- Charged Particles
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Introduction
In our previous article, we explored the problem of dividing a charge q onto two small conducting spheres in such a way that the repulsive force between them is maximized. We found that the charge q should be divided equally between the two spheres, with each sphere carrying a charge of q/2.
In this article, we will address some of the frequently asked questions related to this problem.
Q&A
Q: What is the significance of dividing the charge equally between the two spheres?
A: Dividing the charge equally between the two spheres ensures that the resulting charges are as large as possible, which in turn maximizes the repulsive force between them.
Q: What is the role of Coulomb's Law in this problem?
A: Coulomb's Law is used to calculate the repulsive force between the two spheres. It states that the force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.
Q: How does the distance between the spheres affect the repulsive force?
A: The distance between the spheres has a significant impact on the repulsive force. As the distance between the spheres decreases, the repulsive force increases.
Q: What is the relationship between the charge distribution and the repulsive force?
A: The charge distribution plays a crucial role in determining the repulsive force between the two spheres. By dividing the charge equally between the two spheres, we can maximize the repulsive force.
Q: Can the repulsive force be maximized for any value of distance between the spheres?
A: No, the repulsive force can only be maximized for a specific value of distance between the spheres. This value is determined by the charge distribution and the properties of the spheres.
Q: How does the shape and size of the spheres affect the repulsive force?
A: The shape and size of the spheres can affect the repulsive force, but only to a limited extent. The repulsive force is primarily determined by the charge distribution and the distance between the spheres.
Q: Can the repulsive force be maximized for multiple spheres?
A: Yes, the repulsive force can be maximized for multiple spheres by dividing the charge equally among them.
Conclusion
In conclusion, dividing a charge q onto two small conducting spheres in such a way that the repulsive force between them is maximized is a complex problem that requires a deep understanding of electrostatics. By dividing the charge equally between the two spheres, we can maximize the repulsive force, which is a fundamental principle of electrostatics.
References
- Coulomb, C. A. (1785). Theorie des affections electriques et du magnetisme. Paris: Imprimerie Royale.
- Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press.
Further Reading
- Electromagnetism: Principles and Applications by I. D. Abrahams
- Classical Electrodynamics by J. D. Jackson
Related Topics
- Electrostatics
- Coulomb's Law
- Electric Fields
- Charged Particles