Problem 69 - University Physics with Modern Physics

Question

A charge \(+Q\) is located at the origin, and a charge \(+Q\) is at distance \(d\) away on the \(x\)-axis. Where should a third charge, \(q\), be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

Step-by-Step Solution

Answer

Place -q equidistantly on the y-axis, with magnitude \(|q| = \frac{Qd^2}{2r^2}\).

Step 1: Understand the System

We have two positive charges, both with charge +Q, placed on the x-axis at positions x=0 and x=d. We need to place a third charge, q, such that the system is in equilibrium, meaning the net force on each charge is zero.

Step 2: Consider the Charge Positions

Since the two charges are identical and placed symmetrically around the y-axis, placing the third charge along the x-axis (between or outside the existing charges) would allow us to use symmetry in solving the problem. However, to ensure equilibrium, the forces on each charge must be considered, including their direction.

Step 3: Determine the Placement of the Third Charge

The third charge should not be placed between the two +Q charges because it would be repelled by both, making it impossible for the two charges +Q to reach equilibrium. Instead, place the third charge along the y-axis, equidistant from both +Q charges at the midpoint (which is at x=d/2 on the x-axis, y=some value on the y-axis). This placement utilizes symmetry for equilibrium.

Step 4: Calculate the Required Magnitude and Sign of the Third Charge

Since we are placing the charge equidistant from the two +Q charges on a perpendicular bisector, the third charge, q, must exert forces that counteract the repulsion between the two +Q's. For equilibrium, the forces due to this third charge must provide exactly the repelling force needed. The charge q must be opposite in sign, hence -q. By Coulomb's law, the magnitude of charge q, considering symmetry and equilibrium conditions, can be derived using:\[ k\frac{Q^2}{d^2} = 2\times k\frac{Q|q|}{r^2} \]where r is the distance from each +Q to the position of q. Solving, we find:\[ |q| = \frac{Qd^2}{2r^2} \]

Step 5: Conclude with the Charge's Position and Value

The third charge -q should be placed at the midpoint along the y-axis equidistant to the two +Q charges. The exact position on the y-axis and the magnitude can be calculated based on the calculated distance r, which should satisfy equilibrium in specific scenarios given by additional initial conditions such as system configuration.

Key Concepts

Coulomb's law

Coulomb's law is a crucial principle in electrostatics, governing how charged objects interact with each other. It states that the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
This relationship is mathematically represented as:
\[ F = k \frac{|Q_1 Q_2|}{r^2} \]
where:

• \( F \) is the force between the charges,
• \( k \) is the Coulomb's constant \( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \),
• \( |Q_1| \) and \( |Q_2| \) are the magnitudes of the charges,
• \( r \) is the distance between the charges.

Coulomb's law is incredibly useful for predicting and analyzing how charges will interact. It allows us to calculate exactly how much force each charge will exert on another, helping ensure systems like the one in our problem are in equilibrium. Recognizing that forces exerted by particles on each other must cancel out for equilibrium is key to solving electrostatic problems.

symmetry in physics

The concept of symmetry plays a vital role in understanding physical systems, especially in electrostatics. Symmetry in a system implies that certain properties remain unchanged under specific transformations or positioning.
In the context of our exercise, symmetry allows us to predict the arrangement of charges that will result in equilibrium. By placing a third charge equidistant from two like charges on an axis perpendicular to the line connecting them, we use symmetry to our advantage.
Thanks to symmetry:

• We know forces on objects located symmetrically will often balance out.
• It simplifies complex calculations by reducing the problem's dimensionality.
• Predictive power: Symmetry often helps anticipate the behavior of systems without detailed calculations.

Employing symmetry can be a strategic step in solving physics problems. In electrostatic problems such as this, it helps determine locations and conditions for charges where forces balance, achieving electrostatic equilibrium.

charge interactions

Charge interactions are central to understanding electrostatic equilibrium. In simple terms, charges exert forces on each other. These forces can be attractive or repulsive, depending on the nature of the charges involved.
For like charges (both positive or negative), the interaction is repulsive. Opposite charges attract each other. These interactions form the basis for achieving electrostatic equilibrium, where the net force on each charge in the system is zero.
Key points about charge interactions include:

• The direction of the force will depend on the relative positions of the charges.
• The magnitude of the force is influenced by the distance between the charges – closer charges exert a stronger force on each other.
• In a system with multiple charges, like the one in the exercise, each charge interacts with every other charge, creating a network of forces that must balance for equilibrium.

Understanding these interactions is crucial when calculating where to place a charge so that the system is stable. For the problem at hand, these principles guide where the third charge should be positioned and what its magnitude and sign need to be for maintaining stability in the charge configuration.