Problem 14 - University Physics with Modern Physics

Question

Identical point charges \(q=+5.00 \mu \mathrm{C}\) are placed at opposite corners of a square. The length of each side of the square is 0.200 \(\mathrm{m}\) . A point charge \(q_{0}=-2.00 \mu \mathrm{C}\) is placed at one of the empty comers. How much work is done on \(q_{0}\) by the electric force when \(q_{0}\) is moved to the other empty comer?

Step-by-Step Solution

Answer

The work done is the difference in potential energy: \( (U_f - U_i) \).

Step 1: Understanding the Problem

We have a square with side length of 0.200 m. Two identical positive charges, each of +5.00 μC, are placed at opposite corners. A third point charge of -2.00 μC is initially placed at one of the empty corners. We need to calculate the work done to move this charge to the other empty corner.

Step 2: Find Initial Position Energy

The initial energy is the potential energy due to the interaction of the charge at an empty corner with the two fixed charges. The potential energy between two point charges is given by\[ U = k \frac{q_1 q_2}{r} \]where \(k\) is the Coulomb's constant \(8.99 \times 10^9 \text{ Nm}^2/ ext{C}^2\), \(q_1\) and \(q_2\) are the magnitudes of the charges and \(r\) is the distance between them.

Step 3: Calculate Initial Position Energy

The negative charge \(q_0 = -2.00 \, \mu C\) interacts with each positive charge \(q = +5.00 \, \mu C\) located at a distance of 0.200 m. Thus, the total initial potential energy \(U_i\) is twice the potential energy between one pair:\[ U_i = 2 \cdot \left( k \frac{(+5.00 \times 10^{-6})(-2.00 \times 10^{-6})}{0.200} \right) \]

Step 4: Find Final Position Energy

When moved diagonally, the negative charge \(q_0\) is now \(\sqrt{0.200^2 + 0.200^2} = \sqrt{0.0800} = 0.283 \text{ m}\) away from each positive charge. Calculate the final potential energy \(U_f\), again twice for each pair:\[ U_f = 2 \cdot \left( k \frac{(+5.00 \times 10^{-6})(-2.00 \times 10^{-6})}{0.283} \right) \]

Step 5: Calculate the Work Done

The work done \(W\) is the change in potential energy as the charge moves from its initial to final position:\[ W = U_f - U_i \]

Step 6: Conclusion

Calculate \(U_i\) and \(U_f\) using the provided formulas, compute the change in potential energy and thus the work done on \(q_0\).

Key Concepts

Coulomb's Law

Coulomb's Law is an essential principle used in electrostatics to describe how the electric force between two point charges is calculated. It states that the force between two charges is directly proportional to the product of the absolute values of the charges and inversely proportional to the square of the distance between them. This can be expressed as:- \( F = k \frac{|q_1 q_2|}{r^2} \) where: - \( F \) is the electric force, - \( q_1 \) and \( q_2 \) are the quantities of the charges, - \( r \) is the distance between the charges, and - \( k \) is Coulomb's constant, approximately \(8.99 \times 10^9 \text{ Nm}^2/\text{C}^2\).
This law helps in understanding how forces act between charged objects. Notably, the force is attractive when charges are opposite and repulsive when charges are like. In the given problem, Coulomb’s Law helps determine how much work is needed for moving a charge in relation to others placed in different positions. By evaluating potential energy changes, we understand how movement affects interactions of charged particles.

Point Charges

Point charges are idealized models used in physics to represent charges with negligible size. They simplify calculations by treating the charge as if all of it is located at a single point in space.
In practical applications, this allows us to apply formulas such as those in Coulomb's Law effectively, without worrying about the charge's shape or internal distribution.

• The charges in the given exercise, +5.00 μC and -2.00 μC, are considered point charges.
• Their interactions are modeled and calculated using point charge principles, enabling straightforward computation of electric forces and potential energy changes.

Although idealized, the point charge model is crucial in electrostatics and other fields where electrical interactions are studied.

Electrostatics Problem Solving

Electrostatics involves solving problems related to electric charges at rest. It encompasses understanding forces, fields, and potential energies associated with these charges. Here's how you can effectively approach such problems:
1. **Identify given information and what needs to be determined**: As shown in the example, start by analyzing the initial setup—charges, their positions, and desired calculations, such as work done.
2. **Use appropriate formulas**: Leverage Coulomb's Law for forces, and potential energy equations for energy-related calculations. For point charges, the potential energy \( U \) is given by \( U = k \frac{q_1 q_2}{r} \).
3. **Calculate initial and final conditions**: This often involves finding initial and final potential energies to assess change, as seen in determining work done with the equation \( W = U_f - U_i \).
4. **Combine results to find your answer**: Finally, synthesize your computations to reach the solution, ensuring units are consistent throughout.
By following a structured problem-solving approach, handling electrostatic problems becomes more manageable, enhancing understanding of underlying concepts.