Question

Three particles, each having a charge of \(10 \mu \mathrm{c}\) are placed at the corners of an equilateral triangle of side \(10 \mathrm{~cm}\). The electrostatic potential energy of the system is (Given \(\left.\left[1 /\left(4 \pi \epsilon_{0}\right)\right]=9 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{c}^{2}\right)\) (A) \(100 \mathrm{~J}\) (B) \(27 \mathrm{~J}\) (C) Zero (D) Infinite

Short answer

The total electrostatic potential energy of the system is (B) \(27 J\).

Step-by-step solution

Define the given information

We are given the following information: - Charge of each particle: \(q = 10 \mu C = 10 \times 10^{-6} C\) - Side length of the equilateral triangle: \(d = 10 cm = 0.1 m\) - Coulomb's constant: \(\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 Nm^2/C^2\)

Calculate the electrostatic potential energy between each pair of particles

The formula for the electrostatic potential energy (U) between two particles with charges \(q_1\) and \(q_2\) separated by a distance r is: \[U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}\] In our case, all three particles have the same charge, so \(q_1 = q_2 = q\), and they are separated by the same distance (the side length of the equilateral triangle), so \(r = d\). By applying the formula, we get the potential energy between each pair of particles: \[U_{pair} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{d} = 9 \times 10^9 \frac{(10 \times 10^{-6})^2}{0.1}\]

Calculate the total electrostatic potential energy of the system

As there are 3 pairs of particles, the total electrostatic potential energy of the system (U_total) is obtained by summing the potential energy of each pair: \[U_{total} = 3 \times U_{pair}\] To find the value of \(U_{total}\), we plug the expression for \(U_{pair}\) that we found in step 2: \(U_{total} = 3 \times 9 \times 10^9 \frac{(10 \times 10^{-6})^2}{0.1}\)

Calculate the numerical value of the total electrostatic potential energy

Now, we can calculate the numerical value of the total electrostatic potential energy: \(U_{total} = 3 \times 9 \times 10^9 \frac{(10 \times 10^{-6})^2}{0.1} = 27 J\) So, the correct answer is (B) \(27 J\).

Key concepts

Coulomb's law

Coulomb's law is a fundamental principle used in electromagnetism to calculate the force between two charged objects. Named after Charles-Augustin de Coulomb, this law is pivotal for understanding electrostatic interactions.
Coulomb's law states that the force (F) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:

• Force, \( F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \)

where:

• \( q_1 \) and \( q_2 \) are the magnitudes of the two charges,
• \( r \) is the distance between the charges, and
• \( \frac{1}{4\pi\epsilon_0} \) is Coulomb's constant, which has a value of \( 9 \times 10^9 \, \mathrm{N}\cdot\mathrm{m}^2/\mathrm{C}^2 \).

This law helps us understand how charged objects interact in a non-contact manner, revealing the behavior of forces across space. In exercises such as the one described, Coulomb's law allows us to compute not just forces, but also potential energies, giving deep insights into the energies involved in electrostatic systems.

Equilateral Triangle

An equilateral triangle is a type of polygon that has all three sides of equal length and all three angles equal, typically measuring 60 degrees each. In the context of physics problems, such as the one given, this uniformity provides symmetry that simplifies calculations, especially those involving forces and energies.
Equilateral triangles are often used in problems related to electrostatics due to their geometric properties. With equilateral triangles, each pair of charges in a problem like ours will be equidistant from one another, which allows for uniform calculations and helps in understanding the distribution of forces and potential energy within the system.
For example, when analyzing the electrostatic interaction in an equilateral triangle with point charges at each vertex, such symmetry ensures that the calculations of energy and force required are consistent across each pair of charges, reducing complexity and aiding in comprehending electrostatic potential energy distributions.

Electrostatics Concepts

Electrostatics is the study of charged particles at rest. It involves forces, fields, and potential energies that arise due to static charges. Understanding electrostatics is crucial for solving exercises involving interactions between charged objects.
Here are some core electrostatics concepts relevant to the problem:

• Charge: It is a fundamental property of matter that causes it to experience a force when close to other electrically charged matter.
• Electric Field: A field around a charged particle where a force would be exerted on other charges.
• Electrostatic Potential Energy: This is the energy stored between charged particles due to their positions. It is analogous to gravitational potential energy.
• Superposition Principle: This principle states that the total force or potential energy is the vector sum of individual forces or energies. For multiple charges, like in our problem, we calculate pairwise interactions and sum them up.

By understanding these concepts, one can analyze complex systems involving multiple charges, predict how charges will behave, and compute important quantities like potential energy, as shown in the step-by-step solution provided. Electrostatics forms the backbone of many electronic, chemical, and physical processes, making it an indispensable field of study for students and scientists alike.