Question

Three point charges are positioned on the \(x\) -axis: \(+64.0 \mu \mathrm{C}\) at \(x=0.00 \mathrm{~cm},+80.0 \mu \mathrm{C}\) at \(x=25.0 \mathrm{~cm},\) and \(-160.0 \mu C\) at \(x=50.0 \mathrm{~cm} .\) What is the magnitude of the electrostatic force acting on the \(+64.0-\mu C\) charge?

Short answer

Answer: The magnitude of the electrostatic force acting on the +64.0 µC charge is 6.8 N.

Step-by-step solution

Identify the charges and their positions

We have three point charges: 1. \(q_1 = +64.0 \ \mu C\) at \(x_1=0.00 \ cm\) 2. \(q_2 = +80.0 \ \mu C\) at \(x_2=25.0 \ cm\) 3. \(q_3 = -160.0 \ \mu C\) at \(x_3=50.0 \ cm\) Our goal is to find the magnitude of the electrostatic force acting on the \(q_1\) charge.

Use Coulomb's Law to calculate the forces

Coulomb's Law states that the magnitude of the force between two point charges \(q_1\) and \(q_2\) separated by a distance \(r\) is given by: \(F = k \frac{|q_1 \cdot q_2|}{r^2}\) Where \(k\) is the electrostatic constant, \(k = 8.99 \times 10^9 \ N \cdot m^2 \ / \ C^2\). Now, we will calculate the force between each pair of charges.

Step 2a: Calculate the force between \(q_1\) and \(q_2\)

The distance between \(q_1\) and \(q_2\) is \(r_{12} = x_2 - x_1 = 25.0 \ cm = 0.25 \ m\). Thus, the force between \(q_1\) and \(q_2\) is: \(F_{12} = k \frac{|q_1 \cdot q_2|}{r^2_{12}} = 8.99 \times 10^9 \frac{|(+64 \times 10^{-6}) \cdot (+80 \times 10^{-6})|}{(0.25)^2\ m^2} = 11.5 \ N\) Since both charges are positive, the force is repulsive, so \(F_{12}\) acts to the right.

Step 2b: Calculate the force between \(q_1\) and \(q_3\)

The distance between \(q_1\) and \(q_3\) is \(r_{13} = x_3 - x_1 = 50.0 \ cm = 0.50 \ m\). Thus, the force between \(q_1\) and \(q_3\) is: \(F_{13} = k \frac{|q_1 \cdot q_3|}{r^2_{13}} = 8.99 \times 10^9 \frac{|(+64 \times 10^{-6}) \cdot (-160 \times 10^{-6})|}{(0.50)^2\ m^2} = 18.3 \ N\) Since the charges have opposite signs, the force is attractive, so \(F_{13}\) acts to the left.

Calculate the net force on \(q_1\)

To find the net force acting on the \(+64.0\ \mu C\) charge, we simply add the forces resulting from the \(+80.0\ \mu C\) and the \(-160.0\ \mu C\) charges in terms of their directions. \(F_{net} = F_{13} + (-F_{12}) = 18.3 - 11.5 = 6.8 \ N\) Thus, the magnitude of the electrostatic force acting on the \(+64.0\ \mu C\) charge is \(6.8\ N\).

Key concepts

Coulomb's Law

Understanding Coulomb's Law is essential for calculating the force between two charged objects. Coulomb's Law gives us the formula to determine the force (\( F \)) between two point charges: \[ F = k \frac{|q_1 \cdot q_2|}{r^2} \]where:

• \( k \) is the electrostatic constant, approximately \( 8.99 \times 10^9 \, N \, \cdot \, m^2 \, / \, C^2 \).
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges.
• \( r \) is the distance between the centers of the two charges.

Coulomb's Law tells us the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. This means as the distance increases, the force decreases rapidly. Likewise, larger charges cause a stronger force. Always keep in mind that like charges repel each other, while opposite charges attract.

Point Charges

Point charges refer to charged objects that are so small, we can consider them as being located at a single point for simplicity. This allows us to use Coulomb's Law effectively, as it caters to calculations involving point charges. In practice, point charges are an idealization, but they are extremely useful in theoretical and educational contexts, like in this exercise.
For example, in the given problem, three point charges are situated along the x-axis. When solving problems involving point charges, it’s important to keep track of both their magnitudes and positions to accurately determine distances between them and apply the appropriate calculations. The given positions simplify the problem, as we can quickly establish the distances needed for our calculations. Remember, the interaction between point charges follows fundamental electrostatic principles, but is best understood through the lens of idealized scenarios.

Vector Addition

When dealing with forces in physics, understanding vector addition is crucial because forces have both magnitude and direction. In electrostatics, it's not enough to simply calculate the forces; you need to account for their directions to find the net force.
In our problem, the forces acting on a charge are determined using Coulomb's Law. Once you have these, the next step is to consider their directions:

• Repulsive forces between like charges push them apart.
• Attractive forces between opposite charges pull them together.

To find the net force, one must add these vectorially, considering their directions. For example, if a force acts to the right and another to the left, they need to be subtracted to find the resultant force.
In our exercise example, the repulsive force from one charge is subtracted from the attractive force from another to find the net effect on the selected charge. This illustrates why understanding both magnitude and direction is important for accurately solving electrostatic problems.

Electrostatic Interaction

Electrostatic interaction refers to the forces between charged objects due to their electric charge. These interactions are a fundamental part of physics and are described by electric fields and Coulomb’s Law.
In the context of this exercise, electrostatic interactions define the forces exerted on the point charges by one another. The interactions are dictated by the nature of the charges involved:

• Like charges (e.g., both positive or both negative) exert repulsive forces on each other.
• Opposite charges (one positive, one negative) exert attractive forces on each other.

Electrostatic interactions in this exercise compel us to understand how point charges influence one another.
These forces influence the net force on an individual charge and dictate the behavior of charged particles. Beyond this specific exercise, electrostatic interactions have vast implications, affecting how molecules bond, how electrical devices operate, and much more. A deep understanding of these forces is crucial for anyone looking to explore fields like electrical engineering, chemistry, or applied physics.