Calculate Work Done Assembling Charges At Equilateral Triangle Vertices (PYQ 2020)

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In this detailed article, we will delve into the fascinating realm of electrostatics and tackle a classic problem involving the calculation of work done in assembling point charges. Specifically, we will address the scenario presented in the question: Three point charges, 1µC, -1µC, and 2µC, were initially at an infinite distance apart. Calculate the work done in assembling these charges at the vertices of an equilateral triangle of side 10cm (PYQ 2020). This problem not only tests our understanding of electrostatic potential energy but also highlights the fundamental principles governing the interaction of charges.

Understanding Electrostatic Potential Energy

Before we dive into the solution, it's crucial to grasp the concept of electrostatic potential energy. Electrostatic potential energy is the energy possessed by a system of charges due to their relative positions. It represents the work done in bringing these charges from infinity to their current configuration. The work done in assembling a system of charges is equal to the change in its electrostatic potential energy. This energy arises from the electrostatic forces between the charges, which can be either attractive or repulsive.

When dealing with multiple charges, the total electrostatic potential energy is the sum of the potential energies due to all possible pairs of charges. For two point charges, q1 and q2, separated by a distance r, the electrostatic potential energy (U) is given by:

U = k * (q1 * q2) / r

where k is the electrostatic constant (approximately 8.99 x 10^9 Nm²/C²).

This formula is the cornerstone of our calculation. It tells us that the potential energy is directly proportional to the product of the charges and inversely proportional to the distance between them. A positive potential energy indicates that work needs to be done to bring the charges together (they repel), while a negative potential energy suggests that the charges attract each other and release energy as they come closer.

Problem Setup and Approach

Now, let's apply this concept to the given problem. We have three charges: q1 = 1µC, q2 = -1µC, and q3 = 2µC. These charges are to be assembled at the vertices of an equilateral triangle with sides of 10 cm (0.1 m). Our goal is to calculate the total work done in assembling this configuration, which is equivalent to the final electrostatic potential energy of the system.

To find the total potential energy, we need to consider the potential energy due to each pair of charges: q1 and q2, q1 and q3, and q2 and q3. We will calculate the potential energy for each pair using the formula mentioned above and then sum them up to get the total potential energy. This approach leverages the principle of superposition, which states that the total electrostatic potential energy is the algebraic sum of the individual potential energies.

By breaking down the problem into pairwise interactions, we simplify the calculation and gain a clearer understanding of the contributions from each pair of charges. This methodical approach is essential for solving complex electrostatic problems.

Step-by-Step Calculation of Work Done

Let's proceed with the calculation step by step:

  1. Convert Charges to Coulombs:

    • q1 = 1µC = 1 x 10^-6 C
    • q2 = -1µC = -1 x 10^-6 C
    • q3 = 2µC = 2 x 10^-6 C

    Converting the charges to Coulombs is crucial because the electrostatic constant (k) uses Coulombs as the unit for charge. Consistent units are essential for accurate calculations in physics.

  2. Calculate Potential Energy for Each Pair:

    • U12 (Potential energy between q1 and q2): U12 = k * (q1 * q2) / r = (8.99 x 10^9 Nm²/C²) * (1 x 10^-6 C * -1 x 10^-6 C) / 0.1 m U12 = -0.0899 J

      The negative sign indicates that these charges attract each other, which aligns with the fact that one is positive and the other is negative.

    • U13 (Potential energy between q1 and q3): U13 = k * (q1 * q3) / r = (8.99 x 10^9 Nm²/C²) * (1 x 10^-6 C * 2 x 10^-6 C) / 0.1 m U13 = 0.1798 J

      Here, the positive sign indicates repulsion between the charges, as both are positive.

    • U23 (Potential energy between q2 and q3): U23 = k * (q2 * q3) / r = (8.99 x 10^9 Nm²/C²) * (-1 x 10^-6 C * 2 x 10^-6 C) / 0.1 m U23 = -0.1798 J

      Again, the negative sign signifies attraction between these charges due to their opposite signs.

  3. Calculate Total Potential Energy (Total Work Done):

    • Utotal = U12 + U13 + U23 = -0.0899 J + 0.1798 J - 0.1798 J
    • Utotal = -0.0899 J

    The total potential energy is the sum of the individual potential energies. This gives us the total work done in assembling the charges.

Result and Interpretation

The total work done in assembling the three charges at the vertices of the equilateral triangle is -0.0899 J. The negative sign indicates that the system's potential energy is lower in this configuration than when the charges were infinitely far apart. This means that the attractive forces between the charges dominate the repulsive forces, and the system is more stable in the assembled state. In other words, the system releases energy as the charges are brought together.

This result provides valuable insight into the nature of electrostatic interactions. The balance between attractive and repulsive forces determines the stability and potential energy of a charge configuration. Understanding this balance is fundamental to analyzing various electrostatic phenomena.

Key Takeaways and Significance

This problem serves as an excellent example of how to apply the concept of electrostatic potential energy to calculate the work done in assembling charge configurations. Here are some key takeaways:

  • Electrostatic potential energy is a fundamental concept in electromagnetism, representing the energy stored in a system of charges due to their interactions.
  • The work done in assembling charges is equal to the change in electrostatic potential energy.
  • The potential energy between two point charges is directly proportional to the product of the charges and inversely proportional to the distance between them.
  • The total potential energy of a system of charges is the sum of the potential energies due to all possible pairs of charges.
  • The sign of the potential energy indicates whether the interaction is attractive (negative) or repulsive (positive).
  • This type of problem often appears in introductory physics courses and exams, emphasizing the importance of understanding electrostatic principles.

Furthermore, the principles demonstrated in this problem are widely applicable in various fields, including electronics, materials science, and chemical bonding. For instance, understanding electrostatic interactions is crucial for designing electronic devices, predicting the properties of materials, and explaining the formation of chemical bonds.

Conclusion

In conclusion, we have successfully calculated the work done in assembling three point charges at the vertices of an equilateral triangle. By carefully applying the concept of electrostatic potential energy and breaking down the problem into manageable steps, we arrived at the answer of -0.0899 J. This problem not only reinforces our understanding of electrostatic principles but also highlights the importance of these principles in various scientific and technological applications. Mastering these concepts is essential for any student of physics or engineering and lays the groundwork for further exploration of electromagnetism and its applications.