Question

Two positive point charges of \(12 \mu \mathrm{c}\) and \(8 \mu \mathrm{c}\) are placed \(10 \mathrm{~cm}\) apart in air. The work done to bring them \(4 \mathrm{~cm}\) closer is (A) Zero (B) \(4.8 \mathrm{~J}\) (C) \(3.5 \mathrm{~J}\) (D) \(-5.8 \mathrm{~J}\)

Short answer

The work done to bring the two point charges $4\mathrm{~cm}$ closer is approximately \(5.76 \mathrm{J}\). The given options contain an error, as none of them match the calculated result.

Step-by-step solution

Identify given values and the formula to use

The given values are: \(Q_1 = 12 \mu \mathrm{C} = 12 \times 10^{-6} \mathrm{C}\), \(Q_2 = 8 \mu \mathrm{C} = 8 \times 10^{-6} \mathrm{C}\), initial distance \(r_1 = 10 \mathrm{cm} = 0.1 \mathrm{m}\), and final distance \(r_2 = 6 \mathrm{cm} = 0.06 \mathrm{m}\). We will use the formula for the electrostatic potential energy: \[ U = \frac{kQ_1Q_2}{r} \]

Calculate the initial potential energy

Find the initial potential energy by plugging in the values into the formula: \[ U_1 = \frac{kQ_1Q_2}{r_1} \] \[ U_1 = \frac{(9 \times 10^9 \mathrm{Nm^2C^{-2}})(12 \times 10^{-6} \mathrm{C})(8 \times 10^{-6} \mathrm{C})}{0.1 \mathrm{m}} \] \[ U_1 = 8.64 \mathrm{J} \]

Calculate the final potential energy

Find the final potential energy by plugging in the values into the formula: \[ U_2 = \frac{kQ_1Q_2}{r_2} \] \[ U_2 = \frac{(9 \times 10^9 \mathrm{Nm^2C^{-2}})(12 \times 10^{-6} \mathrm{C})(8 \times 10^{-6} \mathrm{C})}{0.06 \mathrm{m}} \] \[ U_2 = 14.4 \mathrm{J} \]

Calculate the work done

Find the work done by calculating the difference between the final and initial potential energies: \[ W = U_2 - U_1 \] \[ W = 14.4 \mathrm{J} - 8.64 \mathrm{J} \] \[ W = 5.76 \mathrm{J} \] Since the work done is not equal to the given options, we can conclude that the exercise has an error in its options. The correct answer should be approximately \(5.76 \mathrm{J}\).

Key concepts

Electrostatic Potential Energy

Electrostatic potential energy is the energy that is stored due to the positions of charged particles in an electric field. It is an important concept to understand in electrostatics because it describes how charged objects interact at a distance. The formula used to calculate the electrostatic potential energy between two point charges is \[ U = \frac{kQ_1Q_2}{r} \]where:

• \( U \) is the electrostatic potential energy, measured in joules (J),
• \( k \) is the electrostatic constant, approximately \( 9 \times 10^9 \text{Nm}^2\text{C}^{-2} \),
• \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges involved in coulombs (C),
• \( r \) is the distance between the charges in meters (m).

To find the electrostatic potential energy, it's crucial to note how distance influences the energy. As charged particles move closer together, the potential energy increases if the charges are like-signed (both positive or both negative), meaning it takes more energy to come even closer, as shown in our example where moving closer from \( 10 \text{cm} \) to \( 6 \text{cm} \) increased the electrostatic potential energy.

Coulomb's Law

Coulomb's Law is a fundamental principle that describes the force between two point charges. Named after the French physicist Charles-Augustin de Coulomb, the law states that the force (\( F \)) between the charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is given by:\[ F = \frac{kQ_1Q_2}{r^2} \]where:

• \( F \) is the electrostatic force between the charges, measured in newtons (N),
• \( k \) is Coulomb's constant, again approximately \( 9 \times 10^9 \text{Nm}^2\text{C}^{-2} \),
• \( Q_1 \) and \( Q_2 \) are the point charges,
• \( r \) is the distance between the charges.

Essentially, Coulomb’s Law helps us understand how charged particles attract or repel each other. The electrostatic force calculated is a central concept in electrostatics, as it lays the groundwork for understanding how electric fields and potentials work in both simple and complex systems. Recognizing the relationship between force, charge, and distance from this law helps explain the principles calculated in the exercise.

Work Done in Electrostatics

Work done in electrostatics refers to the energy required to move a charge within an electric field. When charges are moved against their natural electric attraction or repulsion, work is performed. This work can be calculated by considering the change in electrostatic potential energy. The formula used is:\[ W = U_2 - U_1 \]where:

• \( W \) is the work done,
• \( U_2 \) is the final potential energy when the charges are at the new distance,
• \( U_1 \) is the initial potential energy when the charges are at their original distance.

In the provided exercise, the work done to bring two charges closer is calculated by finding the difference in potential energy between the two distances. Calculating work done helps quantify the effort needed to overcome the electrical forces involved. It’s significant in various applications, such as in designing electrical systems or understanding energy transformations in electric fields. Just like other forms of work in physics, it’s an indicator of energy transfer, which is indispensable in studying interactions and behaviors of charged objects.