Problem 11 - University Physics with Modern Physics

Question

Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides \(d\). Two of the point charges are identical and have charge \(q\). If zero net work is required to place the three charges at the corners of the triangle, what must the value of the third charge be?

Step-by-Step Solution

Answer

The value of the third charge must be \(-\frac{q}{2}\).

Step 1: Understanding the Work-Energy Principle

The work required to bring charges from infinity to a particular configuration is equal to the potential energy of the final arrangement. If this work is zero, the total electric potential energy of the system is zero.

Step 2: Calculating Potential Energy of the System

Consider three point charges at the corners of an equilateral triangle. The potential energy (\( U \)) is the sum of the potential energy for each pair of charges: \[ U = k \left( \frac{q \cdot q}{d} + \frac{q \cdot Q}{d} + \frac{q \cdot Q}{d} \right), \]where \( k \) is Coulomb's constant, \( q \) is the charge of two identical charges, \( Q \) is the unknown third charge, and \( d \) is the distance between each pair.

Step 3: Setting Total Potential Energy to Zero

Given that the net work is zero, the total potential energy should be equal to zero. So, set \( U = 0 \) and simplify:\[ k \left( \frac{q^2}{d} + \frac{2qQ}{d} \right) = 0. \]

Step 4: Solving for the Unknown Charge

Rearrange the equation to solve for \( Q \):\[ \frac{q^2}{d} + \frac{2qQ}{d} = 0 \]\[ q^2 + 2qQ = 0 \]\[ 2qQ = -q^2 \]\[ Q = -\frac{q^2}{2q} = -\frac{q}{2}. \]

Step 5: Verifying the Solution

Verify that the value \( Q = -\frac{q}{2} \) makes the total potential energy zero: \[ U = k \left( \frac{q^2}{d} + \frac{2q(-\frac{q}{2})}{d} \right) = k \left( \frac{q^2}{d} - \frac{q^2}{d} \right) = 0. \]Thus, the total potential energy is indeed zero, confirming \( Q = -\frac{q}{2} \) as the correct solution.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the force between two charged objects. The law states that the electric 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: \[ F = k \frac{|q_1 q_2|}{r^2} \]where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them. This formula helps us understand how charged particles interact with each other in the universe. Whether it's electrons or protons, Coulomb's Law is essential for calculating the forces that govern their behavior.

• Directly proportional: As the charge magnitudes increase, the force increases.
• Inversely proportional: As the distance between charges increases, the force decreases.

This principle is vital in calculating potential energies and forces in various applications, such as when charges are arranged in triangular configurations.

Point Charges

Point charges are idealized charges that are treated as if they are located at a single point in space. In reality, they have a volume, but for calculations, especially at large distances, treating them as point-like simplifies the math. This simplification is particularly helpful when applying Coulomb's Law to systems of multiple charges, such as those at the corners of a triangle. Distinct examples of point charges include:

• Electrons and protons, which are small enough that they can often be treated as point charges.
• Special cases in physics problems where charges are assumed to be point-like for simplicity.

When analyzing systems with point charges, we consider:

• The individual contributions of each charge to the electric field or potential of the system.
• The interactions between each pair of charges in the system, which are often determined using Coulomb’s Law.

Understanding point charges is crucial for accurately computing the potential energy of assemblies of charges, like those in equilateral triangles.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all sides are of equal length and all angles are equal, typically measuring \( 60^{\circ} \) each. This uniformity is useful in physics because it simplifies the math required to solve problems, like calculating the electric potential energy or forces between point charges placed at its corners.When considering charges at the vertices of an equilateral triangle:

• The symmetry means that the distances between each pair of charges are all equal, making calculations straightforward.
• Applications often involve using symmetry to reduce complex problems to simpler forms.
• Knowing the side length allows for easy application of geometric properties to determine distances and angles needed in calculations.

Using an equilateral triangle for placing point charges not only makes problem-solving more manageable but also exemplifies how geometric configurations influence physical systems. For instance, understanding the properties of these triangles is essential in deriving the electric potential energy of the system or determining unknown charges.