Problem 48 - Essential Calculus Early Transcendentals

Question

Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge \(-1\) at a position \(x\) between them. It follows from Coulomb's Law that the net force acting on the middle particle is $$F(x)=-\frac{k}{x^{2}}+\frac{k}{(x-2)^{2}} \quad 0< x <2$$ where \(k\) is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?

Step-by-Step Solution

Answer

The graph of the net force has an equilibrium point where the net force is zero between 0 and 2, showing where the forces cancel out.

Step 1: Understanding Coulomb's Law

According to Coulomb's Law, the force between two charges is given by \( F = k \frac{{|q_1 q_2|}}{{r^2}} \), where \( k \) is the Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

Step 2: Setting up the Force Equation

Given the charges are \( 1 \) and \(-1\), and they are located at \( 0 \), \( x \), and \( 2 \) on the coordinate line, we express the force as \( F(x) = -\frac{k}{x^2} + \frac{k}{(x-2)^2} \). The first term represents the force exerted on the charge at \( x \) by the charge at \( 0 \), and the second term represents the force exerted by the charge at \( 2 \).

Step 3: Analyzing the Expression for Force

The expression contains two terms: \(-\frac{k}{x^2}\) which is the attractive force from the charge at \( 0 \), and \(\frac{k}{(x-2)^2}\) the repulsive force from the charge at \( 2 \). Their algebraic sum gives the net force on the middle charge at point \( x \).

Step 4: Finding the Sign of the Net Force

To understand the behavior of \( F(x) \), note that the net force depends on the difference \( -\frac{k}{x^2} + \frac{k}{(x-2)^2} \). If \( x \) is closer to \( 0 \), the first term dominates, making the net force negative and pointing left. Conversely, if \( x \) is closer to \( 2 \), the second term dominates, making the net force positive and pointing right.

Step 5: Sketching the Force Graph

Plotting \( F(x) = -\frac{k}{x^2} + \frac{k}{(x-2)^2} \) for \( 0 < x < 2 \): The graph will have an asymptote at \( x = 0 \) and \( x = 2 \) and crosses the x-axis in between. As \( x \) approaches these endpoints, the magnitude of the net force approaches infinity, indicating strong repulsive or attractive forces.

Step 6: Interpreting the Graph

The graph indicates that there is a point where the net force changes sign, suggesting an equilibrium point where the net force is zero. This point divides the region where the force is either attractive or repulsive depending on the proximity to \( 0 \) or \( 2 \).

Key Concepts

Electric Force

The electric force is a fundamental interaction between charged particles. It is shaped by Coulomb's Law, which defines how this force behaves. Essentially, the electric force between two point charges depends on two main factors: the magnitude of the charges and the distance separating them. This relationship is expressed mathematically as \( F = k \frac{{|q_1 q_2|}}{{r^2}} \). Here, \( F \) is the magnitude of the electric force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between these charges. The force is:

• Directly proportional to the product of the charges. This means if the charge magnitude increases, the force strengthens.
• Inversely proportional to the square of the distance between the charges. As they get closer, the force increases significantly.

Electric forces can be

• Attractive if the charges are opposite, such as one positive and one negative.
• Repulsive if both charges have the same sign, either both positive or both negative.

Understanding the nature of electric force is pivotal in analyzing interactions between charged particles.

Charged Particles

In the given exercise, we deal with three charged particles positioned linearly along a coordinate line. They are located at 0, \( x \), and 2. The charges at positions 0 and 2 both possess a charge of \( +1 \), whereas the particle in the middle has a charge of \( -1 \). These particle charges form:

• The charge at position 0 attracts the charge at position \( x \) with a force directed towards the left.
• Meanwhile, the charge at position 2 repels the charge at \( x \) towards the right due to like charges' repulsion.

When interpreting these interactions, remember that

• The attractive force
from the charge at 0 is given by the expression \( -\frac{k}{x^2} \).
• The repulsive force from the charge at 2 is expressed as \( \frac{k}{(x-2)^2} \).

Hence, we derive this net force equation by summing up these two forces. The net force on the middle particle is a balancing act between these repelling and attracting forces, which is influenced by the particle's specific position \( x \) between 0 and 2.

Graph Analysis

Visualizing the net force equation through graph analysis provides a more intuitive understanding of the forces at play between the charged particles. The specific function, \( F(x) = -\frac{k}{x^2} + \frac{k}{(x-2)^2} \), describes how the net force on the middle particle changes as its position \( x \) varies between 0 and 2.The graph of this function:

• Shows asymptotes at the positions 0 and 2. As \( x \) approaches these positions, the force tends towards infinity, indicating extremely strong forces felt by the middle charge at these ends.
• Crosses the x-axis, pinpointing an equilibrium point where the forces balance each other out perfectly.
• Illustrates changes in force direction; to the left of the equilibrium point, the force is negative (leftward), and to the right, it becomes positive (rightward).

Graph analysis not only reveals the magnitude and direction of the net forces at various points but also helps identify the equilibrium where the forces on the middle charge are perfectly balanced. Interpreting such graphs is essential in predicting particle behavior in different positions within any electric field.