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 third charge \(Q\) must be \(-\frac{q}{2}\).

Step 1: Calculate Work for Like Charges

The work required to bring two identical charges \(q\) from infinitely far away to a distance \(d\) apart is given by the formula for electric potential energy between two charges: \[ W_{qq} = \frac{kq^2}{d} \]where \(k\) is Coulomb's constant.

Step 2: Consider Potential Energy of Third Charge

The work done to bring the third charge \(Q\) to the corners with the other two charges \(q\) already placed is \[ 2 \times \frac{kqQ}{d} \]This takes into account the interaction with both of the \(q\) charges.

Step 3: Set Up the Net Work Equation

The total work done should be zero according to the problem statement, so:\[ \frac{kq^2}{d} + 2 \times \frac{kqQ}{d} = 0 \]

Step 4: Solve for the Unknown Charge

Rearrange the equation:\[ \frac{kq^2}{d} + \frac{2kqQ}{d} = 0 \]Simplify:\[ kq^2 + 2kqQ = 0 \]\[ q^2 + 2qQ = 0 \]Solve for \(Q\):\[ Q = -\frac{q}{2} \]

Key Concepts

Point Charges

Point charges are entities with a definite amount of electric charge. They are considered as a single point in space where all the charge is concentrated. In physics problems, point charges are often idealized representations of charged objects.

• Point charges can be positive or negative, with positive charges repelling each other and negative charges doing the same.
• However, a positive charge attracts a negative charge, and vice versa.

In the exercise, we work with three point charges. Two have an equal charge of \(q\) and the third charge, denoted as \(Q\), needs to be found. These charges are placed at the corners of an equilateral triangle. Working with point charges allows us to simplify the complex nature of electrical interactions into precise mathematical equations. Understanding the characteristics of point charges is essential to solving problems involving electric potential energy.

Work and Energy in Physics

Work and energy are key concepts in physics that help describe how forces affect the motion of objects. Work is done when a force causes displacement. It is the product of force and the distance moved along the line of the force's direction.Energy can take many forms, potential energy being one of them. Electric potential energy is the energy stored due to the positions of charged particles. It depends on the configuration of the charges and the distances between them.In the context of the exercise, bringing two charges together requires work. This work is stored as electric potential energy. Calculating this energy for identical point charges \(q\) brought to a distance \(d\) apart plays a crucial part in solving the exercise:\[W_{qq} = \frac{kq^2}{d}\]Then, when another charge \(Q\) is introduced, additional work is required. However, the total work needed to place all charges into position cannot be more than zero, requiring careful consideration of how forces due to each charge act against each other.

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the electric force between two point charges. It states that the magnitude of the electrostatic force attracted by 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 their centers.The formula is:\[F = \frac{k |q_1 q_2|}{r^2}\]

• \(F\) is the magnitude of the force between the charges.
• \(k\) is Coulomb's constant, approximately \(8.9875 \times 10^9 \, \text{N m}^2/\text{C}^2\).
• \(q_1\) and \(q_2\) are the magnitudes of the charges.
• \(r\) is the distance between the charges.

In our exercise, Coulomb's Law helps us derive the expression for the electric potential energy, which is a critical step in determining the configuration of the charges such that no net work is required. By analyzing the force balance and potential energy, we can determine that the third charge \(Q\) must be \(-\frac{q}{2}\) to balance the system and satisfy the condition of zero total work.