Problem 25 - Physics

Question

Charges \(q_{1}\) and \(q_{2}\) are fixed in place, \(q_{2}\) being located at a distance \(d\) to the right of \(\mathrm{q}_{1} . \mathrm{A}\) third charge \(q_{3}\) is then fixed to the line joining \(q_{1}\) and \(q_{2}\) at a distance \(d\) to the right of \(q_{2}\). The third charge is chosen so the potential energy of the group is zero; that is, the potential energy has the same value as that of the three charges when they are widely separated. Determine the value for \(q_{3}\), assuming that (a) \(q_{1}=q_{2}=q\) and (b) \(q_{1}=q\) and \(q_{2}=-q .\) Express your answers in terms of \(q\)

Step-by-Step Solution

Answer

(a) \( q_3 = -\frac{2q}{3} \), (b) \( q_3 = -q \).

Step 1: Understand the potential energy equation

The potential energy between two point charges is given by the formula: \( U = k \frac{q_1 q_2}{r} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them. Our goal is to set the total potential energy of the system to zero.

Step 2: Express the total potential energy for the charges

The total potential energy for the system consisting of three charges along a straight line can be expressed as: \( U_{total} = k\left( \frac{q_1 q_2}{d} + \frac{q_1 q_3}{2d} + \frac{q_2 q_3}{d} \right) = 0 \). Here, \( d \) is the distance between \( q_1 \) and \( q_2 \), and \( 2d \) is the distance between \( q_1 \) and \( q_3 \).

Step 3: Solve for \( q_3 \) when \( q_1 = q_2 = q \)

Substitute \( q_1 = q_2 = q \) into the expression for \( U_{total} \): \[ k \left( \frac{q \cdot q}{d} + \frac{q \cdot q_3}{2d} + \frac{q \cdot q_3}{d} \right) = 0 \]. Simplify this to:\[ \frac{kq^2}{d} + \frac{kqq_3}{2d} + \frac{kqq_3}{d} = 0 \], which simplifies further to:\[ \frac{kq^2}{d} + \frac{3kqq_3}{2d} = 0 \].Solving for \( q_3 \), we get: \( q_3 = -\frac{2q}{3} \).

Step 4: Solve for \( q_3 \) when \( q_1 = q \) and \( q_2 = -q \)

Substitute \( q_1 = q \) and \( q_2 = -q \) into the expression for \( U_{total} \):\[ k \left( \frac{q(-q)}{d} + \frac{q q_3}{2d} - \frac{q q_3}{d} \right) = 0 \]. Simplify this to:\[ -\frac{kq^2}{d} + \frac{kq q_3}{2d} - \frac{kq q_3}{d} = 0 \], which simplifies to:\[ -\frac{kq^2}{d} + \frac{-kq q_3}{2d} = 0 \]. Solving for \( q_3 \), we get: \( q_3 = -q \).

Key Concepts

Electrostatics

Electrostatics is the branch of physics that deals with the study of stationary or slow-moving electric charges. The core idea here is understanding how charges interact with each other when they aren't in motion. Think about it like playing a game of chess, where you carefully analyze each piece's static position to determine their influence on the board.

In electrostatics, we deal with concepts such as electric forces, electric fields, and electric potential energy. Understanding these can help us predict how charged objects will behave when brought close to one another. In our exercise, the static electric forces and resulting potential energy are the main areas of interest. These forces between charges are what keep them bound or repel them when no other movement occurs.

Key aspects include:

• Electric Charge: The fundamental property of matter that causes it to experience a force when close to other electric charges.
• Electric Field: A region around a charged object where other charges will experience a force.
• Electric Potential Energy: The energy stored in a system of charged objects due to their positions relative to each other.

By mastering these concepts, understanding the behavior of charges in scenarios like our exercise becomes more intuitive.

Coulomb's Law

Coulomb's Law is essential for understanding the forces between electric charges. It’s like the fundamental rulebook for predicting how charges will interact at any given time.

The law states that the electric force (\( F \)) between two point charges (\( q_1 \) and \( q_2 \)) is directly proportional to the product of their charges and inversely proportional to the square of the distance (\( r \)) between them:\[ F = k \frac{q_1 q_2}{r^2} \]where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \) N m²/C².

This equation resembles gravity's law but for electric charges. It tells us that:

• Like charges repel: Positive-positive or negative-negative interactions will push away from each other.
• Opposite charges attract: A positive and a negative charge will pull towards each other.
• Distance matters: The force magnitude decreases quickly as the distance increases.

In the original problem, understanding this interaction helps determine the necessary third charge (\( q_3 \)) to achieve zero net potential energy. By following Coulomb's Law, we predict how the potential energy will change as introductory arrangement varies.

Potential Energy Equation

The potential energy equation is the backbone for analyzing systems of electric charges. It's how we quantify the energy stored due to the positions and interactions of charges.

For two charges, the electric potential energy (\( U \)) is given by:\[ U = k \frac{q_1 q_2}{r} \]where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charge magnitudes, and \( r \) is the distance between them. This formula highlights the strength and nature of interactions:

• Positive energy: When both charges are of the same sign, resulting in repulsion.
• Negative energy: When charges differ in sign, leading to attraction.

To find the value of \( q_3 \) in various scenarios mentioned in the exercise, the total potential energy must be zero:\[ U_{total} = k \left( \frac{q_1 q_2}{d} + \frac{q_1 q_3}{2d} + \frac{q_2 q_3}{d} \right) = 0 \]The \(q_3\) calculation requires substituting known values, simplifying this equation, and solving for \( q_3 \). By setting the arrangement so that the energy sums to zero, \( q_3 \) complements the initial charges' potential energy, balancing the system's dynamics.