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 \(+5.4 \times 10^{-8} \)J. When the second charge is moved to point \(b\), the electric force on the charge does \(-1.9 \times 10^{-8}\) 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 \(3.5 \times 10^{-8}\) J.

Step 1: Understand the Original Electric Potential Energy

The electric potential energy of two point charges is determined by their charges and the distance between them. Initially, when the second charge \(q_2\) is at point \(a\), the electric potential energy \(U_a\) is given as \(+5.4 \times 10^{-8}\) J.

Step 2: Understand Work Done by Electric Force

When work is done on a charge, it means energy has been transferred. In this case, work done on the charge \(q_2\) as it moves from point \(a\) to point \(b\) is \(-1.9 \times 10^{-8}\) J. The negative sign indicates that the electric force did work against the movement, suggesting a reduction in potential energy.

Step 3: Calculate New Electric Potential Energy at Point b

The change in electric potential energy \(\Delta U\) = Work done by the electric force, so: \(\Delta U = U_b - U_a = -1.9 \times 10^{-8}\) J. Solving for \(U_b\), we get: \(U_b = U_a + (-1.9 \times 10^{-8}\) J \(= 5.4 \times 10^{-8}\) J - \(1.9 \times 10^{-8}\) J \(= 3.5 \times 10^{-8}\) J.

Key Concepts

Point Charges

Point charges are idealized charges that are imaginary points in space where the whole charge is concentrated. They are useful in simplifying problems where charge distribution is otherwise complex. Because they are points, they have no volume, surface area, or any spatial extent.
Each point charge creates an electric field around itself, influencing other charges nearby. The strength of interaction between point charges depends on:

• The magnitude of each charge
• The distance between them

The formula used to calculate the electric potential energy between two point charges is:\[ U = k \frac{q_1 q_2}{r} \]where:

• \( U \) is the electric potential energy
• \( k \) is Coulomb's constant \(8.99 \times 10^9 \text{ Nm}^2/\text{C}^2\)
• \( q_1 \) and \( q_2 \) are the magnitudes of the point charges
• \( r \) is the distance between the two charges

Electric Force

Electric force is the force of repulsion or attraction between two charged objects. It is one of the fundamental forces of physics and arises due to electric charges interacting with each other. Like electric potential energy, the force also depends on the magnitude of the charges and the distance between them. The electric force equation is given by Coulomb's law:\[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the electric force
• \( q_1 \) and \( q_2 \) are the point charges
• \( r \) is the distance separating the charges
• \( k \) is Coulomb's constant

The electric force is directional:

• Like charges repel each other.
• Unlike charges attract each other.

In our exercise, while moving the charge from point \( a \) to point \( b \), the electric force exerted by the stationary point charge adjusts the potential energy of the system.

Work Done

The concept of work done in the context of electric forces relates to energy transfer. Work done is a measure of energy that is transferred as a charge moves through an electric field. When a charge is moved against the direction of the electric force, energy is required, leading to positive work done.
Conversely, if the movement is along the force, work obtained from the system is negative, implying energy is released. In our exercise, the work performed by the electric force while moving a charge from point \( a \) to point \( b \) was \(-1.9 \times 10^{-8}\) J. The negative value indicates that the system released energy as the charge moved between the points.
The relationship between work done (\( W \)) and the change in electric potential energy (\( \Delta U \)) can be expressed as:\[ \Delta U = W \]This means that the change in electric potential energy is equal to the work performed by or against the electric force. Hence, computing work done aids in understanding how electric potential energy is altered over the course of a charge's movement.