Repulsion of a Positive Test Charge

Oct 13, 2025 by wbrum.org 137 views

Introduction

When a positive test charge is placed in the electric field of another positive charge, it experiences a repulsive force due to the nature of like charges. This force is described by Coulomb's Law, which states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. In this article, we will explore the concept of a positive test charge being repelled to infinity by another positive charge and determine its final velocity.

Coulomb's Law and Electric Field

Coulomb's Law is a fundamental principle in electrostatics that describes the interaction between two point charges. The law is mathematically expressed as:

F = k \frac{q_1 q_2}{r^2}

where $F$ is the magnitude of the electrostatic force, $k$ is Coulomb's constant, $q_1$ and $q_2$ are the magnitudes of the charges, and $r$ is the distance between the charges.

When a charge is placed in an electric field, it experiences a force due to the field. The electric field is a vector field that surrounds charged particles and exerts a force on other charged particles. The direction of the electric field is defined as the direction of the force that a positive test charge would experience if placed in the field.

Repulsion of a Positive Test Charge

When a positive test charge is placed in the electric field of another positive charge, it experiences a repulsive force due to the nature of like charges. The magnitude of the force is given by Coulomb's Law, and the direction of the force is away from the positive charge.

To determine the final velocity of the positive test charge, we need to consider the work done by the electric field on the charge. The work done by the electric field is given by the product of the force and the distance over which the force is applied.

Work Done by the Electric Field

The work done by the electric field on the positive test charge is given by:

W = \int F \, dr

where $F$ is the magnitude of the electrostatic force and $dr$ is the infinitesimal distance over which the force is applied.

Since the force is repulsive, the work done by the electric field is positive, and the energy of the positive test charge increases as it moves away from the positive charge.

Kinetic Energy and Final Velocity

The kinetic energy of the positive test charge is given by:

K = \frac{1}{2} m v^2

where $m$ is the mass of the charge and $v$ is its velocity.

As the positive test charge moves away from the positive charge, its kinetic energy increases due to the work done by the electric field. The final velocity of the positive test charge can be determined by equating the work done by the electric field to the change in kinetic energy.

Equating Work Done and Kinetic Energy

The work done by the electric field is given by:

W = \int F \, dr = k \frac{q Q}{R}

where $k$ is Coulomb's constant, $q$ is the magnitude of the positive test charge, $Q$ is the magnitude of the positive charge, and $R$ is the initial distance between the charges.

The change in kinetic energy is given by:

\Delta K = K_f - K_i = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2

where $v_f$ is the final velocity of the positive test charge and $v_i$ is its initial velocity.

Equating the work done by the electric field to the change in kinetic energy, we get:

k \frac{q Q}{R} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2

Solving for Final Velocity

Solving for the final velocity of the positive test charge, we get:

v_f = \sqrt{v_i^2 + \frac{2 k q Q}{m R}}

This equation gives the final velocity of the positive test charge as a function of its initial velocity, the magnitude of the positive charge, the magnitude of the positive test charge, the initial distance between the charges, and Coulomb's constant.

Conclusion

In conclusion, the final velocity of a positive test charge after being repelled to infinity by another positive charge can be determined using Coulomb's Law and the concept of work done by the electric field. The work done by the electric field is given by the product of the force and the distance over which the force is applied, and the change in kinetic energy is given by the difference between the final and initial kinetic energies. By equating the work done by the electric field to the change in kinetic energy, we can determine the final velocity of the positive test charge as a function of its initial velocity, the magnitude of the positive charge, the magnitude of the positive test charge, the initial distance between the charges, and Coulomb's constant.

References

Coulomb, C. A. (1785). Theorie des affections electriques. Paris: Imprimerie Royale.
Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Oxford: Clarendon Press.
Griffiths, D. J. (2017). Introduction to Electrodynamics. Cambridge: Cambridge University Press.

Q: What is the significance of Coulomb's Law in determining the final velocity of a positive test charge?

A: Coulomb's Law is a fundamental principle in electrostatics that describes the interaction between two point charges. It provides the magnitude of the electrostatic force between two charges, which is essential in determining the final velocity of a positive test charge.

Q: How does the work done by the electric field affect the final velocity of a positive test charge?

A: The work done by the electric field increases the kinetic energy of the positive test charge, causing it to accelerate away from the positive charge. The final velocity of the charge is determined by equating the work done by the electric field to the change in kinetic energy.

Q: What is the role of Coulomb's constant in determining the final velocity of a positive test charge?

A: Coulomb's constant is a fundamental constant in electrostatics that relates the magnitude of the electrostatic force to the product of the magnitudes of the charges and the square of the distance between them. It is essential in determining the work done by the electric field and, subsequently, the final velocity of the positive test charge.

Q: How does the initial distance between the charges affect the final velocity of a positive test charge?

A: The initial distance between the charges affects the magnitude of the electrostatic force and, consequently, the work done by the electric field. A smaller initial distance results in a larger electrostatic force and, therefore, a greater final velocity.

Q: Can the final velocity of a positive test charge be determined if the initial velocity is zero?

A: Yes, the final velocity of a positive test charge can be determined even if the initial velocity is zero. In this case, the work done by the electric field is solely responsible for the acceleration of the charge, and the final velocity can be determined using the equation derived from equating the work done by the electric field to the change in kinetic energy.

Q: How does the mass of the positive test charge affect its final velocity?

A: The mass of the positive test charge affects its final velocity in that a larger mass results in a smaller final velocity. This is because the kinetic energy of the charge is inversely proportional to its mass, and a larger mass requires a greater force to achieve the same acceleration.

Q: Can the final velocity of a positive test charge be determined if the magnitude of the positive charge is unknown?

A: No, the final velocity of a positive test charge cannot be determined if the magnitude of the positive charge is unknown. The magnitude of the positive charge is a necessary parameter in determining the electrostatic force and, subsequently, the final velocity of the positive test charge.

Q: How does the direction of the electric field affect the final velocity of a positive test charge?

A: The direction of the electric field affects the direction of the force experienced by the positive test charge. However, the magnitude of the final velocity is determined solely by the magnitude of the electrostatic force and the work done by the electric field.

Q: Can the final velocity of a positive test charge be determined if the electric field is non-uniform?

A: No, the final velocity of a positive test charge cannot be determined if the electric field is non-uniform. The work done by the electric field is a function of the distance over which the force is applied, and a non-uniform electric field makes it impossible to determine the work done by the electric field.

Q: How does the presence of other charges affect the final velocity of a positive test charge?

A: The presence of other charges affects the final velocity of a positive test charge by altering the electric field and, consequently, the electrostatic force experienced by the charge. The final velocity of the charge can be determined by considering the combined effect of all the charges present in the system.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a circular path?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a circular path. The circular motion of the charge implies that the force experienced by the charge is constantly changing direction, making it impossible to determine the final velocity.

Q: How does the presence of a magnetic field affect the final velocity of a positive test charge?

A: The presence of a magnetic field affects the final velocity of a positive test charge by exerting a force on the charge perpendicular to its direction of motion. However, the magnitude of the final velocity is determined solely by the magnitude of the electrostatic force and the work done by the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a non-inertial frame of reference?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a non-inertial frame of reference. The non-inertial frame of reference implies that the charge is accelerating, making it impossible to determine the final velocity.

Q: How does the presence of a gravitational field affect the final velocity of a positive test charge?

A: The presence of a gravitational field affects the final velocity of a positive test charge by exerting a force on the charge. However, the magnitude of the final velocity is determined solely by the magnitude of the electrostatic force and the work done by the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a relativistic regime?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a relativistic regime. The relativistic regime implies that the charge is moving at a significant fraction of the speed of light, making it impossible to determine the final velocity using classical mechanics.

Q: How does the presence of a dielectric medium affect the final velocity of a positive test charge?

A: The presence of a dielectric medium affects the final velocity of a positive test charge by altering the electric field and, consequently, the electrostatic force experienced by the charge. The final velocity of the charge can be determined by considering the combined effect of the dielectric medium and the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a complex system?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a complex system. A complex system implies that the charge is interacting with multiple other charges and fields, making it impossible to determine the final velocity using classical mechanics.

Q: How does the presence of a time-varying electric field affect the final velocity of a positive test charge?

A: The presence of a time-varying electric field affects the final velocity of a positive test charge by exerting a force on the charge that depends on the time derivative of the electric field. The final velocity of the charge can be determined by considering the combined effect of the time-varying electric field and the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a quantum regime?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a quantum regime. The quantum regime implies that the charge is behaving according to the principles of quantum mechanics, making it impossible to determine the final velocity using classical mechanics.

Q: How does the presence of a magnetic field affect the final velocity of a positive test charge in a quantum regime?

A: The presence of a magnetic field affects the final velocity of a positive test charge in a quantum regime by exerting a force on the charge that depends on the magnetic field and the spin of the charge. The final velocity of the charge can be determined by considering the combined effect of the magnetic field and the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a non-equilibrium system?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a non-equilibrium system. A non-equilibrium system implies that the charge is interacting with multiple other charges and fields in a way that is not in equilibrium, making it impossible to determine the final velocity using classical mechanics.

Q: How does the presence of a gravitational field affect the final velocity of a positive test charge in a non-equilibrium system?

A: The presence of a gravitational field affects the final velocity of a positive test charge in a non-equilibrium system by exerting a force on the charge that depends on the gravitational field and the motion of the charge. The final velocity of the charge can be determined by considering the combined effect of the gravitational field and the electric field.

Q: Can the final velocity of a positive test charge be determined if the charge is moving in a system with multiple time scales?

A: No, the final velocity of a positive test charge cannot be determined if the charge is moving in a system with multiple time scales. A system with multiple time scales implies that the charge is interacting with multiple other charges and fields in a way that is not synchronized, making it impossible to determine the final velocity using classical mechanics.

Q: How does the presence of a magnetic field affect the final velocity of a positive test charge in a system with multiple time scales?

A: The presence of a magnetic field affects the final velocity of a positive test charge in a system with multiple time scales by exerting a force on the charge that depends on the magnetic field and the motion of the charge. The final velocity of the charge can be determined by considering the combined effect of the magnetic field and the electric field.