Problem 2 - University Physics with Modern Physics

Question

A point charge \(q_{1}\) is held stationary at the origin. A second charge \(q_{2}\) is placed at point \(a,\) and the electric potential energy of the pair of charges is \(+5.4 \times 10^{-8} \mathrm{J}\) . When the second charge is moved to point \(b\) , the electric force on the charge does \(-1.9 \times 10^{-8} \mathrm{J}\) of work. What is the electric potential energy of the pair of charges when the second charge is at point \(b ?\)

Step-by-Step Solution

Answer

The electric potential energy at point b is \(3.5 \times 10^{-8} \mathrm{J}\).

Step 1: Understand the Concept

Electric potential energy between two point charges is expressed as the work needed to move one of the charges from infinity to its current position. The change in potential energy is the work done by the electric force when moving the charge.

Step 2: Identify the Given Values

The electric potential energy at point \(a\) is \(+5.4 \times 10^{-8} \mathrm{J}\), and the work done by the electric force in moving the charge from \(a\) to \(b\) is \(-1.9 \times 10^{-8} \mathrm{J}\).

Step 3: Calculate Potential Energy at Point b

Use the formula: \( \Delta U = U_b - U_a = -W \) Given \( \Delta U = -1.9 \times 10^{-8} \mathrm{J}\) and \(U_a = +5.4 \times 10^{-8} \mathrm{J}\), solve for \(U_b\): \[ U_b = U_a + (-W) = 5.4 \times 10^{-8} \mathrm{J} + (-1.9 \times 10^{-8} \mathrm{J}) \] \[ U_b = 5.4 \times 10^{-8} \mathrm{J} - 1.9 \times 10^{-8} \mathrm{J} = 3.5 \times 10^{-8} \mathrm{J} \]

Step 4: Finalize the Solution

The electric potential energy of the pair of charges when the second charge is at point \(b\) is \(3.5 \times 10^{-8} \mathrm{J}\).

Key Concepts

Point Charges

Understanding the concept of point charges is fundamental in the study of electric potential energy. In physics, point charges refer to charged particles that are so small, they can be treated as if they occupy a single point in space. This simplifies calculations because it allows us to focus on the effects of electric forces without worrying about the shape or size of the source charges.
Point charges interact with each other through the electric force, a fundamental interaction in nature. The strength of this force is governed by Coulomb's law, which states that the electric 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 law allows us to predict how charges will affect each other over various distances, helping to calculate their potential energy.
Consider that in our scenario, we have two point charges: \( q_1 \), which is stationary at the origin, and \( q_2 \), which moves between two points, namely \( a \) and \( b \). Understanding these interactions is key to solving problems related to electric potential energy.

Work Done by Electric Force

In the world of electrostatics, work done by the electric force is a critical concept, as it directly influences potential energy. When a charge moves, the electric force can do work on this charge. For instance, moving from point \( a \) to point \( b \) in an electric field involves changes in the potential energy due to work being performed.
The amount of work done by the electric force on a charge can be described by the equation: \[ W = - (U_b - U_a) \]where \( W \) is the work done, and \( U_a \) and \( U_b \) are the electric potential energies at points \( a \) and \( b \) respectively. A negative work value indicates that energy was extracted from the system as the charge moves.
In the exercise scenario, the work done when charge \( q_2 \) moves from point \( a \) to point \( b \) is \( -1.9 \times 10^{-8} \text{ J} \). This computation reveals how the electric force interacts with the charge as it moves in the field created by \( q_1 \).

Change in Potential Energy

The change in electric potential energy is a direct consequence of work done by the electric forces and provides insight into the behavior of charged particles in electric fields. This change is crucial because it helps us determine the energy transformations occurring as the charge moves.
To calculate the change in potential energy when a charge moves from point \( a \) to point \( b \), we utilize the equation:\[ \Delta U = U_b - U_a = -W \]where \( \Delta U \) represents the change in potential energy. This equation shows the relationship between work done and energy change, making it easier to compute new energy states given initial conditions.
For example, in the original exercise, we use the known values of the potential energy at point \( a \) and the work done by the electric force to find the potential energy at point \( b \). This gives the result \( 3.5 \times 10^{-8} \text{ J} \). The clearer understanding of how work contributes to energy changes is pivotal for solving electric potential energy problems efficiently.