Question

Identical point charges \(Q\) are placed at each of the four corners of a rectangle measuring \(2.0 \mathrm{~m}\) by \(3.0 \mathrm{~m} .\) If \(Q=32 \mu C,\) what is the magnitude of the electrostatic force on any one of the charges?

Short answer

Answer: To find the net electrostatic force acting on one of the charges, we need to follow these steps: 1. Identify the forces acting on the charge due to the other charges. 2. Calculate the magnitudes of these forces using Coulomb's law. 3. Resolve the diagonal force into its components along the sides of the rectangle. 4. Add the horizontal and vertical components of the forces separately. 5. Calculate the net force acting on the charge as the square root of the sum of the squares of the total horizontal and vertical components. Using the given values, find the magnitudes of the forces, resolve the diagonal force into components, and calculate the net force acting on the charge.

Step-by-step solution

Identify the forces acting on a charge

We have one charge at each corner of the rectangle. We need to find the electrostatic force acting on one charge, let's say the one at the bottom-left corner. There are three forces acting on this charge due to the other three charges: 1. The force from the charge at the bottom-right corner, horizontally. 2. The force from the charge at the top-left corner, vertically. 3. The force from the charge at the top-right corner, diagonally.

Calculate the forces from individual charges

To calculate the forces from the individual charges, we can use Coulomb's law: \(F=\frac{kq_1q_2}{r^2}\). Here, \(k = 8.99\times10^9 \mathrm{Nm^2C^{-2}}\), \(q_1 = Q\), and \(q_2 = Q\), for all the cases. Let the distance between the charges at the bottom-left and bottom-right corners be \(r_1 = 2.0\mathrm{m}\), and the distance between the charges at the bottom-left and top-left corners be \(r_2 = 3.0\mathrm{m}\). The diagonal distance (\(r_3\)) between the charges at the bottom-left and top-right corners can be found using the Pythagorean theorem: \(r_3 = \sqrt{r_1^2 + r_2^2}\).

Find the magnitudes of the forces

Calculate the magnitudes of the forces acting on the charge at the bottom-left corner due to other charges using Coulomb's law: 1. Force due to the charge at bottom-right corner: \(F_{1} = \frac{kQ^2}{r_1^2}\). 2. Force due to the charge at the top-left corner: \(F_{2} = \frac{kQ^2}{r_2^2}\). 3. Force due to the charge at the top-right corner: \(F_{3} = \frac{kQ^2}{r_3^2}\).

Resolve the diagonal force into components

To find the net force acting on the charge, we need to resolve the diagonal force (\(F_{3}\)) into its components along the sides of the rectangle. Using the triangle that consists of \(r_1, r_2\), and \(r_3\), we can find the angles and then calculate the force components: Horizontal component of \(F_{3}\): \(F_{3x} = F_{3}\cos\theta\). Vertical component of \(F_{3}\): \(F_{3y} = F_{3}\sin\theta\). Here, \(\theta = \arctan\left(\frac{r_2}{r_1}\right)\).

Calculate the net force acting on the charge

Add the horizontal components and vertical components of the forces separately, and find the magnitude of the net force acting on the charge: Horizontal components: \(F_{1}\) and \(F_{3x}\). Vertical components: \(F_{2}\) and \(F_{3y}\). Net force acting on the charge: \(F_\text{net} = \sqrt{(F_{1} + F_{3x})^2 + (F_{2} + F_{3y})^2}\). Now, plug in the values of \(Q, r_1, r_2\), and find \(r_3\), then calculate the magnitudes of forces \(F_{1}, F_{2}\), and \(F_{3}\). Afterward, resolve \(F_{3}\) into its components, and finally calculate the net force acting on the charge.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the electrostatic force between two point charges. Named after Charles-Augustin de Coulomb, who first published the idea in the 18th century, this law is essential for calculating the magnitude of the electrostatic force in physics problems.

The law can be stated mathematically as: \[ F = k \frac{|q_1 q_2|}{r^2} \],where:

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

The force is attractive if the charges are of opposite signs and repulsive if they are like-charged. In the problem provided, because each point charge is identical and positively charged, the force between any two charges will be repulsive.

To apply Coulomb's Law to our rectangle problem, we calculate the force between the charge of interest and each of the other charges separately, using their distance apart as \(r\). This step-by-step application lays the groundwork for understanding how these individual forces combine to produce the net force on a charge.

Superposition Principle

The Superposition Principle is a key concept in physics that allows us to analyze the combined effect of multiple forces acting on a charge. According to this principle, when several charges are present, the total force on any one charge is the vector sum of the forces exerted on that charge by each of the other charges individually. This principle applies remarkably well to electrostatic forces, which are central in our rectangle problem.

Using the Superposition Principle, we consider the electrostatic forces calculated individually from each charge on the target charge we are analyzing. In our case, this means assessing the force contributions from each of the three other charges in the rectangle setup. The total force is the sum of all these forces, taking into account both magnitude and direction.

The vector nature of these forces is crucial. This means we can't simply add the numbers; we must consider the directional components. This is where vector resolution becomes significant, and it is the next concept we will discuss. Applying the Superposition Principle, we simplify complex interactions into manageable calculations.

Vector Resolution

Vector resolution is the process of breaking down a vector into its components. When dealing with forces in physics, particularly electrostatic forces, it's often necessary to resolve a force that's not aligned with the standard axes into horizontal and vertical components. This is done using trigonometric functions based on the geometry of the system.

In our rectangle problem, we have a force that acts diagonally, \(F_3\). To use this force in our calculations effectively, we need to resolve it along the horizontal and vertical axes. The technique involves finding the angle \(\theta\) that the force makes with the horizontal, which can be calculated using the inverse tangent function, \(\arctan\), and the ratios of the rectangle's sides.

Once \(\theta\) is known, the components of the diagonal force are found as follows:

• Horizontal component: \(F_{3x} = F_{3}\cos\theta\)
• Vertical component: \(F_{3y} = F_{3}\sin\theta\)

With these components, we can add them to the other horizontal and vertical forces from the other charges. The net force is then calculated by forming a right triangle with the total horizontal and vertical forces, and applying the Pythagorean theorem to find the magnitude of the resultant vector.