There Are Three Point Charges Of Equal magnitude Q Placed At The Three Corners Of A right Angle Triangle, As Shown In Fig. 15.2. AB = AC. What Is The Magnitude And Direction Of the Force Exerted On – Q
Introduction
When dealing with multiple point charges, the forces exerted on each charge can be complex to calculate. However, by breaking down the problem into smaller components and using the principles of superposition, we can determine the net force acting on each charge. In this scenario, we have three point charges of equal magnitude q placed at the three corners of a right angle triangle. The goal is to find the magnitude and direction of the force exerted on the charge – q.
Understanding the Problem
To solve this problem, we need to consider the forces exerted on the charge – q by the other two charges. Since the charges are placed at the corners of a right angle triangle, we can use the properties of right triangles to simplify the calculations. The charges are placed at points A, B, and C, with AB = AC. We will use the coordinates of the charges to calculate the forces exerted on – q.
Calculating the Forces
Let's consider the forces exerted on – q by the charges q at points A and B. We can use the formula for the force exerted by a point charge on another point charge:
F = k * (q1 * q2) / r^2
where F is the force, k is Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Since the charges are placed at the corners of a right angle triangle, we can use the Pythagorean theorem to find the distance between the charges:
r^2 = AB^2 + BC^2
We can simplify the calculations by using the coordinates of the charges. Let's assume that the charge – q is placed at the origin (0, 0), and the charges q are placed at points (a, 0) and (0, b).
Calculating the Force Exerted by Charge q at Point A
The force exerted by the charge q at point A on the charge – q is given by:
F_A = k * (q * -q) / (a^2 + b^2)
Since the charges have equal magnitude, we can simplify the expression:
F_A = -k * q^2 / (a^2 + b^2)
Calculating the Force Exerted by Charge q at Point B
The force exerted by the charge q at point B on the charge – q is given by:
F_B = k * (q * -q) / (a^2 + b^2)
Since the charges have equal magnitude, we can simplify the expression:
F_B = -k * q^2 / (a^2 + b^2)
Calculating the Net Force
The net force exerted on the charge – q is the sum of the forces exerted by the charges q at points A and B:
F_net = F_A + F_B
Since the forces are equal and opposite, we can simplify the expression:
F_net = -2 * k * q^2 / (a^2 + b^2)
Determining the Magnitude and Direction of the Net Force
The magnitude of the net force is given by:
F_net = 2 * k * q^2 / (a^2 + b^2)
The direction of the net force can be determined by considering the signs of the forces. Since the forces exerted by the charges q at points A and B are equal and opposite, the net force will be directed along the line joining the charges.
Conclusion
In conclusion, the magnitude and direction of the force exerted on the charge – q can be determined by calculating the forces exerted by the charges q at points A and B. By using the principles of superposition and the properties of right triangles, we can simplify the calculations and determine the net force acting on the charge – q.
Final Answer
The magnitude of the net force is given by:
F_net = 2 * k * q^2 / (a^2 + b^2)
The direction of the net force is along the line joining the charges.
References
- [1] Griffiths, D. J. (2017). Introduction to Electrodynamics. Pearson Education.
- [2] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
Discussion
This problem is a classic example of how to calculate the forces exerted on a charge by multiple point charges. By breaking down the problem into smaller components and using the principles of superposition, we can determine the net force acting on each charge. The properties of right triangles can be used to simplify the calculations and determine the magnitude and direction of the net force.
Related Problems
- [1] Three point charges of equal magnitude q are placed at the corners of an equilateral triangle. What is the magnitude and direction of the force exerted on the charge – q?
- [2] Two point charges of equal magnitude q are placed at points A and B. What is the magnitude and direction of the force exerted on the charge – q at point C?
See Also
- [1] Coulomb's Law
- [2] Electric Field
- [3] Superposition Principle
Introduction
In our previous article, we discussed how to calculate the forces exerted on a charge by multiple point charges. We used the principles of superposition and the properties of right triangles to simplify the calculations and determine the net force acting on each charge. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the difference between the force exerted by a single point charge and the force exerted by multiple point charges?
A: The force exerted by a single point charge is given by Coulomb's Law, which states that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. When multiple point charges are present, the forces exerted by each charge must be added together using the principle of superposition.
Q: How do I determine the direction of the net force exerted on a charge by multiple point charges?
A: To determine the direction of the net force, you must consider the signs of the forces exerted by each charge. If the forces are equal and opposite, the net force will be directed along the line joining the charges. If the forces are not equal and opposite, the net force will be directed along the line joining the charges, but with a different magnitude.
Q: Can I use the same formula to calculate the force exerted by multiple point charges as I would for a single point charge?
A: No, you cannot use the same formula to calculate the force exerted by multiple point charges as you would for a single point charge. The formula for the force exerted by a single point charge is given by Coulomb's Law, but when multiple point charges are present, the forces exerted by each charge must be added together using the principle of superposition.
Q: What is the significance of the distance between charges in calculating the force exerted by multiple point charges?
A: The distance between charges is crucial in calculating the force exerted by multiple point charges. The force between two charges is inversely proportional to the square of the distance between them, so the distance between charges must be taken into account when calculating the net force.
Q: Can I use the same method to calculate the force exerted by multiple point charges as I would for a different type of charge, such as a dipole or a quadrupole?
A: No, you cannot use the same method to calculate the force exerted by multiple point charges as you would for a different type of charge, such as a dipole or a quadrupole. Each type of charge has its own unique properties and requires a different method for calculating the force exerted by multiple charges.
Q: What is the relationship between the force exerted by multiple point charges and the electric field?
A: The force exerted by multiple point charges is related to the electric field. The electric field is a vector field that describes the force exerted by a charge on other charges. When multiple point charges are present, the electric field is the sum of the electric fields due to each charge.
Q: Can I use the same method to calculate the force exerted by multiple point charges as I would for a different type of charge, such as a charged sphere or a charged cylinder?
A: No, you cannot use the same method to calculate the force exerted by multiple point charges as you would for a different type of charge, such as a charged sphere or a charged cylinder. Each type of charge has its own unique properties and requires a different method for calculating the force exerted by multiple charges.
Q: What is the significance of the principle of superposition in calculating the force exerted by multiple point charges?
A: The principle of superposition is crucial in calculating the force exerted by multiple point charges. It states that the total force exerted on a charge by multiple charges is the sum of the forces exerted by each charge individually.
Q: Can I use the same method to calculate the force exerted by multiple point charges as I would for a different type of charge, such as a charged wire or a charged plate?
A: No, you cannot use the same method to calculate the force exerted by multiple point charges as you would for a different type of charge, such as a charged wire or a charged plate. Each type of charge has its own unique properties and requires a different method for calculating the force exerted by multiple charges.
Conclusion
In conclusion, calculating the forces exerted on a charge by multiple point charges requires a deep understanding of the principles of superposition and the properties of right triangles. By using the correct formulas and methods, you can determine the net force acting on each charge and understand the behavior of charges in different situations.
Final Answer
The force exerted by multiple point charges is given by the principle of superposition, which states that the total force exerted on a charge by multiple charges is the sum of the forces exerted by each charge individually.
References
- [1] Griffiths, D. J. (2017). Introduction to Electrodynamics. Pearson Education.
- [2] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
Discussion
This article provides a comprehensive overview of the forces exerted on a charge by multiple point charges. By understanding the principles of superposition and the properties of right triangles, you can calculate the net force acting on each charge and understand the behavior of charges in different situations.
Related Problems
- [1] Three point charges of equal magnitude q are placed at the corners of an equilateral triangle. What is the magnitude and direction of the force exerted on the charge – q?
- [2] Two point charges of equal magnitude q are placed at points A and B. What is the magnitude and direction of the force exerted on the charge – q at point C?
See Also
- [1] Coulomb's Law
- [2] Electric Field
- [3] Superposition Principle