Question

Two identically charged particles separated by a distance of \(1.00 \mathrm{~m}\) repel each other with a force of \(1.00 \mathrm{~N}\). What is the magnitude of the charges?

Short answer

Answer: The magnitude of the charges is approximately \(3.36 \times 10^{-6} \mathrm{C}\).

Step-by-step solution

Coulomb's Law formula

Coulomb's Law states that the electrostatic force between two charged particles is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. The formula for Coulomb's Law is: \(F = k\frac{|q_1q_2|}{r^2}\) where \(F\) is the force between the charges, \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between them, and \(k\) is the electrostatic constant, which has a value of \(8.9875 \times 10^9 \mathrm{Nm^2C^{-2}}\).

Identify given values

We are given the following values: \(F = 1.00 \mathrm{~N}\) \(r = 1.00 \mathrm{~m}\) And since the charges are identical and repel each other, we can say: \(q_1 = q_2 = q\)

Substitute values into the equation

Now, we can substitute these values into the Coulomb's Law equation and solve for the charge \(q\): \(1.00 \mathrm{~N} = 8.9875 \times 10^9 \mathrm{Nm^2C^{-2}} \frac{q^2}{(1.00 \mathrm{~m})^2}\)

Solve for the charge \(q\)

Solving for the charge \(q\), we get: \(q^2 = \frac{1.00 \mathrm{~N} \times (1.00 \mathrm{~m})^2}{8.9875 \times 10^9 \mathrm{Nm^2C^{-2}}}\) \(q^2 = \frac{1.00}{8.9875 \times 10^9}\) \(q = \sqrt{\frac{1.00}{8.9875 \times 10^9}}\) \(q \approx \pm 3.36 \times 10^{-6} \mathrm{C}\) Since the charges are repelling each other, we choose the positive value: \(q = 3.36 \times 10^{-6} \mathrm{C}\) So, the magnitude of the charges is \(3.36 \times 10^{-6} \mathrm{C}\).

Key concepts

Electrostatic Force

Understanding the concept of electrostatic force is crucial when studying the interactions between electrically charged particles. Put simply, this is the force that charged particles exert on each other. It can either be attractive or repulsive depending on whether the charges are of opposite or similar nature, respectively. The principle at work here is similar to the way magnets behave: opposite charges attract, like charges repel.

For example, in the exercise given, two identically charged particles are repelling each other with a force of 1 Newton. What's happening at the microscopic level is an interaction defined by the properties of the charges and the distance between them. This basic understanding helps in visualizing why charged particles behave the way they do and is fundamental to several fields, including electrical engineering and physics.

Electric Charge

The term 'electric charge' is pretty common, but what does it actually entail? Imagine it as an intrinsic property of particles that dictates how they will interact within an electric field. Charges come in two types: positive and negative. Similar charges repel each other, while opposite charges attract.

In the context of our exercise, each particle has a charge, and since they are repelling, we know the charges have the same sign. The charge of particles is measured in coulombs (C), a standard unit in the metric system. So when we find that each particle has a charge of roughly 3.36 x 10^-6 C, we have quantified the amount of 'electricity' they carry. Understanding the concept of electric charge is vital for grasping not only electrostatics but also broader topics in electromagnetism.

Coulomb's Constant

Coulomb's constant, denoted as 'k' in Coulomb's Law, is a value that helps us quantify the electrostatic force between two charges. Its value is approximately 8.9875 x 10^9 Nm^2C^-2. The constant works as a proportionality factor—when you multiply it by the product of the two charges and then divide by the square of the distance between them, you get the electrostatic force.

In our original problem, this constant is crucial in calculating the exact charge of the particles. It essentially sets the scale of the force based on the charges and distance involved. The existence of such a constant illustrates the predictability of physical laws and allows us to apply mathematical models to physical situations with high precision. Coulomb's constant is derived from empirical observations and is an essential component of electrostatic calculations.