A Charge Of 2 10^-9C Is Placed At The Origin, And Another Charge Of 4 10^-9C Is Placed At X = 1.5m. This scenario presents a classic problem in electrostatics, involving the interaction of point charges and the principles governing Coulomb's law. Understanding how these charges influence each other, the nature of the forces involved, and the resulting electric fields is fundamental in physics, particularly in the study of electric forces and potentials. This article explores the detailed analysis of this setup, including calculations, concepts, and real-world applications, providing a comprehensive guide for students, educators, and enthusiasts alike.
Introduction to Electric Charges and Coulomb's Law
Electric charges are intrinsic properties of particles that cause them to exert forces on one another. These forces can be attractive or repulsive depending on the types of charges involved. Coulomb's law provides the quantitative basis for understanding these interactions.What Are Electric Charges?
Electric charges are fundamental properties of matter, classified as positive or negative. Like charges repel each other, while opposite charges attract. The magnitude of the charge influences the strength of the electrostatic force between particles.Coulomb's Law Explained
Coulomb's law states that the magnitude of the electrostatic force \( F \) between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them:\[
F = ke \frac{|q1 q_2|}{r^2}
\]
Where:
- \( F \) is the magnitude of the force between the charges,
- \( k_e \) is Coulomb's constant (\( 8.9875 \times 10^9 \, \mathrm{Nm^2/C^2} \)),
- \( q1, q2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.
This law is fundamental for analyzing electrostatic problems involving multiple charges.
Understanding the Problem Setup
In our scenario, we have two point charges:- A charge of \( 2 \times 10^{-9} \, \mathrm{C} \) located at the origin (\( x = 0 \)),
- A charge of \( 4 \times 10^{-9} \, \mathrm{C} \) located at \( x = 1.5\, \mathrm{m} \).
Visual Representation of the Setup
To better understand the problem, consider a one-dimensional coordinate system:- At \( x=0 \), the first charge \( q_1 = 2 \times 10^{-9} \, \mathrm{C} \),
- At \( x=1.5\, \mathrm{m} \), the second charge \( q_2 = 4 \times 10^{-9} \, \mathrm{C} \).
Calculating Electric Fields
Electric fields are vector quantities that represent the force per unit positive charge experienced at a point in space due to other charges.Electric Field Due to a Point Charge
The electric field \( E \) at a distance \( r \) from a point charge \( q \) is given by:\[
E = k_e \frac{|q|}{r^2}
\]
Direction:
- Radially outward from a positive charge,
- Radially inward toward a negative charge.
Electric Field at a Point Due to Both Charges
When multiple charges are present, the net electric field at a point is the vector sum of the individual fields from each charge.
Step-by-step calculation:
- Identify the points of interest (e.g., at various positions along the line).
- Calculate the distance from each charge to the point.
- Determine the magnitude and direction of the electric field due to each charge.
- Add vectorially to find the net electric field.
Force Between the Charges
Since the charges are fixed at specific points, the force on each due to the other can be calculated directly using Coulomb's law.
Force on \( q_1 \):
\[
F{12} = ke \frac{|q1 q2|}{r^2}
\]
where \( r = 1.5\, \mathrm{m} \).
Force on \( q_2 \):
\[
F{21} = F{12}
\]
by Newton's Third Law.
Direction:
- If both charges are positive, they repel each other.
- If one is negative, the force becomes attractive.
In this case, both are positive, so they repel, and the force acts along the line connecting them, pushing each away.
Calculating Electric Potential
Electric potential at a point due to a point charge is given by:\[
V = k_e \frac{q}{r}
\]
The total potential at a point is the algebraic sum of potentials due to each charge.
Potential Energy of the System
The electrostatic potential energy \( U \) of two point charges is:\[
U = ke \frac{q1 q_2}{r}
\]
This represents the energy stored due to the configuration of the charges.
Applications and Significance of Electrostatic Calculations
Understanding the interaction between these charges has practical applications across various fields:- Design of Capacitors
- Electrostatic Shielding
- Understanding Atomic and Molecular Interactions
- Electronics and Circuit Design
Step-by-Step Calculation Example
Let's perform a sample calculation: the force between the two charges.Given:
- \( q_1 = 2 \times 10^{-9} \, \mathrm{C} \),
- \( q_2 = 4 \times 10^{-9} \, \mathrm{C} \),
- \( r = 1.5\, \mathrm{m} \),
- \( k_e = 8.9875 \times 10^9 \, \mathrm{Nm^2/C^2} \).
Calculations:
\[
F = ke \frac{|q1 q_2|}{r^2} = (8.9875 \times 10^9) \times \frac{(2 \times 10^{-9})(4 \times 10^{-9})}{(1.5)^2}
\]
\[
F = (8.9875 \times 10^9) \times \frac{8 \times 10^{-18}}{2.25}
\]
\[
F = (8.9875 \times 10^9) \times 3.555 \times 10^{-18}
\]
\[
F \approx 3.195 \times 10^{-8} \, \mathrm{N}
\]
This is the magnitude of the force with which the two charges repel each other.
Conclusion
The interaction of the two charges placed at specific positions along a line exemplifies fundamental principles of electrostatics. By applying Coulomb's law, calculating electric fields, and understanding potential energy, one can analyze the forces and potentials involved in such systems. These concepts are foundational to many technological applications, from designing electronic components to understanding atomic interactions. Mastery of these calculations not only enhances comprehension of classical physics but also paves the way for innovations in science and engineering.Additional Resources for Further Learning
- Textbooks on Electromagnetism (e.g., "Introduction to Electrodynamics" by David J. Griffiths)
- Online simulations (e.g., PhET Electric Field & Potential Simulations)
- Educational videos on Coulomb's Law and electric fields
- Practice problems on electrostatics for skill reinforcement
