Problem 21 - College Physics

Question

\(\bullet$$\bullet\) Two point charges are located on the \(y\) axis as \(\mathrm{fol}\) lows: charge \(q_{1}=-1.50 \mathrm{nC}\) at \(y=-0.600 \mathrm{m},\) and charge \(q_{2}=+3.20 \mathrm{nC}\) at the origin \((y=0) .\) What is the net force (magnitude and direction) exerted by these two charges on a third charge \(a_{3}=+5,00\) nC located at \(y=-0.400 \mathrm{m} ?\)

Step-by-Step Solution

Answer

The net force magnitude is \( F_{net} \) (upward direction towards charge \( q_2 \)).

Step 1: Understanding the Problem

We have three point charges. Charge \( q_1 = -1.50 \text{ nC} \) is located at \( y = -0.600 \text{ m} \), charge \( q_2 = +3.20 \text{ nC} \) is at \( y = 0 \), and charge \( q_3 = +5.00 \text{ nC} \) is at \( y = -0.400 \text{ m} \). We need to calculate the net electric force on \( q_3 \), considering the forces from \( q_1 \) and \( q_2 \). Only the \( y \)-component direction and Coulomb's law are relevant here due to alignment on the \( y \) axis.

Step 2: Calculate Force due to Charge q1

Using Coulomb's law, the force between \( q_1 \) and \( q_3 \) is given by: \[ F_{13} = k \frac{|q_1 q_3|}{(y_3-y_1)^2} \]Where \( k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \). Substitute values: \( q_1 = -1.50 \text{ nC}, q_3 = +5.00 \text{ nC} \), and the distance is \( 0.200 \text{ m} \) (\(-0.400 \)m - \(-0.600\)m). Thus, \[ F_{13} = 8.99 \times 10^9 \frac{|-1.50 \times 10^{-9} \cdot 5.00 \times 10^{-9}|}{0.200^2} \]Calculate \( F_{13} \). The direction of this force is towards \( q_1 \) (at \( y=-0.600 \text{ m} \)), because \( q_1 \) is negative and \( q_3 \) is positive.

Step 3: Calculate Force due to Charge q2

The force between \( q_2 \) and \( q_3 \) is: \[ F_{23} = k \frac{|q_2 q_3|}{(y_3-y_2)^2} \]Substitute values: \( q_2 = +3.20 \text{ nC}, q_3 = +5.00 \text{ nC} \), and the distance is \( 0.400 \text{ m} \) (\(-0.400\)m - \(0\)m). Thus, \[ F_{23} = 8.99 \times 10^9 \frac{|3.20 \times 10^{-9} \cdot 5.00 \times 10^{-9}|}{0.400^2} \]Calculate \( F_{23} \). The direction of this force is away from \( q_2 \) (at \( y=0 \)), because both \( q_2 \) and \( q_3 \) are positive.

Step 4: Determine the Net Force

As calculated, \( F_{13} \) acts towards negative \( y \) (down) and \( F_{23} \) acts towards positive \( y \) (up). The net force \( F_{net} \) can be found by vector sum:\[ F_{net} = F_{23} - F_{13} \]Evaluate with your calculated values for \( F_{13} \) and \( F_{23} \). This will give both the magnitude and the direction of the net force.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of electrostatics that helps us calculate the electric force between two point charges. It tells us how these charges interact based on their magnitudes and the distance between them. The formula for Coulomb's Law is given by \[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the electric force
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \)
• \( q_1 \) and \( q_2 \) are the magnitudes of the two charges
• \( r \) is the distance between the charges

Coulomb's Law shows that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The force is attractive if the charges are of opposite signs and repulsive if they have the same sign.

Electric Force

Electric force is the interaction between electrically charged objects. It is a non-contact force, meaning the objects don't need to touch to exert forces on each other. In the exercise, we calculate the electric force exerted on a charge due to other charges using Coulomb's Law.
Electric force has both magnitude and direction. The direction depends on the nature of the charges:

• If the charges are opposite, the force is attractive, pulling the charges towards each other.
• If the charges are the same, the force is repulsive, pushing the charges apart.

Understanding the direction is crucial for correctly determining the net force when multiple charges are involved. In problems set on the same axis, like this one on the y-axis, the forces may simply add or subtract based on their directions.

Point Charges

Point charges are objects treated as if all of their electric charge is concentrated at a single point. This simplification allows us to easily calculate forces between charges as if they have no size. In practice, electrons, protons, and other fundamental particles are point charges, while larger charged objects can sometimes be approximated as point charges for simplification.
In this exercise, we deal with three point charges. Each charge exerts a force on the other due to their electric fields. Point charges make it straightforward to apply Coulomb's Law because calculations for force depend solely on the distance between these points and the magnitude of their charges, without concern for their shapes or sizes.

Vector Addition

When dealing with multiple forces, such as in this electrostatics problem, understanding vector addition is essential. Each force not only has a magnitude but also a direction, making them vector quantities. To find the net force, we sum these vectors using vector addition rules.
In our example, the forces exerted on the point charge from each of the other two charges have specific directions:

• The force from charge \( q_1 \) is directed towards \( q_1 \) itself because \( q_3 \) feels an attractive force.
• The force from charge \( q_2 \) is directed away, as both \( q_2 \) and \( q_3 \) are positive and repel each other.

To find the net force, we subtract the force vector directed downwards from the vector directed upwards because they act along the same line. This gives us both the magnitude and the direction of the net force acting on the charge.