(b) What is the change in potential energy if the distance separating the two electrons is increased to 1.0 nm?

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Identify the formula for the potential energy between two charges: \(U = \frac{k \cdot q_1 \cdot q_2}{r}\), where \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them.

Recognize that the charges involved are electrons, each with a charge of \(-1.6 \times 10^{-19}\) C.

Substitute the known values into the formula: \(k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2\), \(q_1 = q_2 = -1.6 \times 10^{-19}\) C, and \(r = 1.0 \times 10^{-9}\) m.

Calculate the potential energy using the formula, keeping in mind that the product of two negative charges is positive, which affects the sign of the potential energy.

Discuss how increasing the distance between the charges affects the potential energy, noting that as \(r\) increases, the magnitude of \(U\) decreases, indicating a decrease in potential energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged particles. It states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is essential for understanding how the potential energy changes as the distance between charged particles, like electrons, varies.

Potential Energy in Electrostatics

In electrostatics, the potential energy (U) between two point charges is given by the formula U = k * (q1 * q2) / r, where k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between them. As the distance increases, the potential energy decreases for like charges, indicating that they become less repulsive as they move apart. This concept is crucial for calculating the change in potential energy when the distance between two electrons is altered.

Change in Potential Energy

The change in potential energy (ΔU) when the distance between two charges changes can be calculated by finding the potential energy at the initial and final distances and taking the difference. For two electrons, as the distance increases from a smaller value to 1.0 nm, the change in potential energy will reflect the work done against the electrostatic force. Understanding this change is key to answering questions about energy transformations in electrostatic systems.