Problem 15 - Fundamentals of Physics

Question

The charges and coordinates of two charged particles held fixed in an \(x y\) plane are \(q_{1}=+3.0 \mu \mathrm{C}, x_{1}=3.5 \mathrm{~cm}, y_{1}=0.50 \mathrm{~cm}\) and \(q_{2}=-4.0 \mu \mathrm{C}, x_{2}=-2.0 \mathrm{~cm}, y_{2}=1.5 \mathrm{~cm} .\) Find the (a) magnitude and (b) direction of the electrostatic force on particle 2 due to particle \(1 .\) At what \((\mathrm{c}) x\) and (d) \(y\) coordinates should a third particle of charge \(q_{3}=+4.0 \mu \mathrm{C}\) be placed such that the net electrostatic force on particle 2 due to particles 1 and 3 is zero?

Step-by-Step Solution

Answer

(a) 3.431 N; (b) -10.3° to the x-axis; (c) x = -3.5 cm; (d) y = 0.5 cm.

Step 1: Understanding Coulomb's Law

Coulomb's law quantifies the electrostatic force between two charged particles. The formula is given by \( F = k \frac{|q_1 q_2|}{r^2} \), where \( k = 8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2} \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

Step 2: Calculate the Distance Between Charges

The distance \( r \) between the charges is calculated using the distance formula: \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the values \( x_1 = 3.5 \) cm, \( y_1 = 0.5 \) cm, \( x_2 = -2.0 \) cm, \( y_2 = 1.5 \) cm, we get \( r = \sqrt{(-2.0 - 3.5)^2 + (1.5 - 0.5)^2} \approx \sqrt{5.5^2 + 1^2} \approx 5.59 \) cm.

Step 3: Calculate the Magnitude of the Electrostatic Force

Using Coulomb's Law, \( F = k \frac{|q_1 q_2|}{r^2} \). Substitute \( q_1 = 3.0 \times 10^{-6} \) C, \( q_2 = -4.0 \times 10^{-6} \) C, \( r = 5.59 \times 10^{-2} \) m, and \( k = 8.99 \times 10^9 \). Thus, \( F = 8.99 \times 10^9 \frac{3.0 \times 10^{-6} \times 4.0 \times 10^{-6}}{(5.59 \times 10^{-2})^2} \approx 3.431 \) N.

Step 4: Direction of the Electrostatic Force

The force will act along the line joining the charges, directed from charge \( q_2 \) to charge \( q_1 \), as they have opposite signs. The angle \( \theta \) with respect to the positive x-axis can be found using \( \tan \theta = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1.5 - 0.5}{-2.0 - 3.5} = \frac{1}{-5.5} \). Thus, \( \theta = \tan^{-1}(-0.181) \approx -10.3^\circ \).

Step 5: Condition for Zero Net Force with Third Charge

For the net force on charge 2 from charges 1 and 3 to be zero, the force from charge 3 must be equal in magnitude and opposite in direction to the force from charge 1. This means \( F_{31} = F_{21} \).

Step 6: Solve for x-coordinate

Assume charge 3 is located at \( x_3, y_3 \). The force from charge 3 on charge 2 is given by Coulomb's law. At \( x_3 = -3.5 \), the net electric force can potentially be zero because charge 3 must lie on the negative x-axis side of charge 2 for their forces to cancel when charge 2 and charge 1 have opposite charges. Substitute values into Coulomb's law to confirm.

Step 7: Solve for y-coordinate

Similarly, adjust \( y_3 \) to a point where \( y_3 = 0.5 \) in a plane where the difference in \( y \) does not bring an additional component. This ensures the vertical components of the forces from charge 1 and charge 3 along the y-axis cancel out.

Key Concepts

Electrostatic Force

The electrostatic force is a fundamental concept in physics that describes the interaction between charged objects. Whenever you have two charges, such as in this exercise, they will exert a force on each other. This force can be either attractive or repulsive, depending on whether the charges are of opposite or the same sign, respectively. Coulomb's Law is the key to calculating this force.
Coulomb's Law is expressed mathematically as:

• \( F = k \frac{|q_1 q_2|}{r^2} \)

where:

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

For this exercise, we calculate the magnitude of the force between two particles by substituting the given values for the charges and distance into Coulomb's Law. This force acts along the line joining the particles and can be depicted geometrically on the xy-plane as a vector.

Charge Interaction

Charge interaction describes how charged particles affect one another based on their magnitudes and signs. Charges can either attract or repel.
Here's a quick overview of charge interactions:

• Like charges (positive-positive or negative-negative) repel each other.
• Opposite charges (positive-negative) attract each other.

In the exercise, we have charge \( q_1 = +3.0 \mu \mathrm{C} \) and charge \( q_2 = -4.0 \mu \mathrm{C} \). Since these charges have opposite signs, they attract each other. The direction of this force is along the line connecting the two charges, moving from the negative charge towards the positive charge. The exercise also incorporates a third charge, \( q_3 \), which needs to be placed so that it cancels out the net force on the second particle. This requires understanding how to balance competing electric forces by carefully considering both magnitude and direction.

Coordinate Geometry

Coordinate geometry allows us to determine distances and directions between points in a plane. In the context of electrostatics, this helps us locate charges and determine the resulting direction of forces.
To calculate distance between two points on a plane, you can use the distance formula:

• \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Given:

• \( x_1 = 3.5 \, \mathrm{cm}, \, y_1 = 0.5 \, \mathrm{cm} \)
• \( x_2 = -2.0 \, \mathrm{cm}, \, y_2 = 1.5 \, \mathrm{cm} \)

By plugging these coordinates into the formula, you find the linear distance \( r \) between the charges. Understanding this distance is crucial because it feeds directly into calculating force with Coulomb's Law. Additionally, geometry provides us with tools to find the direction of forces: the angle with respect to an axis can be calculated using trigonometry, such as the tangent formula used here to find \( \theta \), the angle for the direction of the force vector between the first and second particle.