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 Initial and Final Conditions

We are given the electric potential energy at point \( a \) as \(+5.4 \times 10^{-8} \; \mathrm{J} \). When the second charge \( q_2 \) is moved to point \( b \), the work done by the electric force is \(-1.9 \times 10^{-8} \; \mathrm{J} \). We need to find the new electric potential energy when the charge is at point \( b \).

Step 2: Use Work-Energy Principle

The work done by the electric force when moving a charge from one point to another is equal to the change in electric potential energy of the system. This can be expressed as:\[ W = U_b - U_a \]where \(W\) is the work done (\(-1.9 \times 10^{-8} \; \mathrm{J}\)), \(U_a\) is the initial electric potential energy at point \( a \) (\(+5.4 \times 10^{-8} \; \mathrm{J}\)), and \(U_b\) is the electric potential energy at point \( b \), which we need to find.

Step 3: Rearrange the Formula

We rearrange the equation from Step 2 to solve for \( U_b \):\[ U_b = W + U_a \]Here, we add the work done by the force to the initial electric potential energy to find the potential energy at point \( b \).

Step 4: Substitute the Known Values

Substitute the values into the rearranged formula:\[ U_b = -1.9 \times 10^{-8} \; \mathrm{J} + 5.4 \times 10^{-8} \; \mathrm{J} \]

Step 5: Calculate the Result

Calculate the electric potential energy at point \( b \):\[ U_b = (5.4 - 1.9) \times 10^{-8} \; \mathrm{J} = 3.5 \times 10^{-8} \; \mathrm{J} \]

Key Concepts

Work-Energy Principle

The Work-Energy Principle is a fundamental concept in physics that relates the work done by forces to the change in energy of a system. Specifically for electric forces, when a charge moves in an electric field, the work done by the electric force will result in a change in the system's electric potential energy. This principle is expressed mathematically as:

• \[ W = U_b - U_a \]

This equation indicates that the work done \( W \) by the electric force during the movement from one point \( a \) to another point \( b \) is equal to the difference in electric potential energies \( U_b \) and \( U_a \).
Let's break this down further:

• "Work done" refers to how much energy the electric force transfers as it moves the charge.
• The term \( U_a \) is the electric potential energy at the initial point \( a \), while \( U_b \) is the final energy at point \( b \).

This principle is pivotal because it helps you calculate unknown variables, like determining the new electric potential energy at point \( b \) when the work done is known.

Point Charge Interaction

Point charge interaction is a concept describing how individual charges exert forces on one another. This is typically visualized in problems involving multiple charges at various positions.
When two point charges are involved, each charge generates its own electric field. The interaction between these fields and charges results in a force exerted on each charge by the other. The force and resulting potential energy depend on several factors:

• The magnitude of each charge: Larger charges produce stronger fields and exert more potent forces.
• The distance between charges: The electric force decreases as the distance between the charges increases, following an inverse square law.
• The nature of the charges: Like charges repel, and opposite charges attract, according to Coulomb's law.

In our exercise, the point charge \( q_1 \) is stationary at the origin, influencing the moving charge \( q_2 \). Understanding point charge interaction is crucial for solving problems involving electric forces and potential energy changes as charges are repositioned.

Electric Force

Electric force is the fundamental force exerted by electric fields on charges. It plays a critical role in determining how charges interact over distances. The electric force can be calculated by Coulomb's law, which states that the force \( F \) between two point charges is:

• Directly proportional to the product of the magnitudes of the charges \( q_1 \) and \( q_2 \).
• Inversely proportional to the square of the distance \( r \) between the charges.
• Described by the formula: \[ F = k\frac{|q_1q_2|}{r^2} \]

where \( k \) is the Coulomb's constant \( 8.99 \times 10^9 \, \mathrm{N\cdot m^2/C^2} \).
Electric forces can either be attractive or repulsive:

• Two like charges (both positive or both negative) will repel each other.
• Two unlike charges (one positive, one negative) will attract each other.

The work performed by these forces during charge movement leads to changes in electric potential energy, as seen in the provided exercise. Understanding this interaction through electric force is key to analyzing systems with multiple charges.