### Application of Coulomb's Law with a Third ChargeWhen a third charge, \( Q_3 \), is introduced into the system, the following configuration arises:#### Diagram Description- **Charges and Positions:** - \( Q_1 \) = +6 μC at the origin. - \( Q_2 \) = +3 μC located 4 meters to the right of \( Q_1 \) on the x-axis. - \( Q_3 \) is located 3 meters above \( Q_1 \) on the y-axis.- **Force Vector and Angle:** - The electric force \( F_1 \) on \( Q_1 \) is acting towards the upper left, at a 45° angle to the vertical.#### Problem StatementA. The electric force on \( Q_1 \) is now acting to the upper left, at a 45° angle to the vertical (as shown in the diagram). Calculate the unknown charge \( Q_3 \).B. Using your answer in part (A), determine the electric force acting on \( Q_2 \). You may answer in vector component form (i.e., \( \mathbf{F_2} = (F_{x}, F_{y}) \)).### Detailed Analysis**Diagram Explanation:**- **\( Q_1 \)** is at the origin.- **\( Q_2 \)** is 4 meters along the x-axis from \( Q_1 \).- **\( Q_3 \)** is 3 meters along the y-axis from \( Q_1 \).- The force \( F_1 \) affects \( Q_1 \) at a 45° angle to the vertical axis heading towards the upper left.The problem involves using Coulomb’s Law and the principles of vector addition to find the unknown charge \( Q_3 \) based on the described electric forces, and then determining the resulting force on \( Q_2 \).### Step-by-Step Solution Approach**Part A:**1. **Identify Individual Forces Contribution:** - Calculate the force due to \( Q_2 \) on \( Q_1 \), \( \mathbf{F}_{12} \). - Calculate the force due to \( Q_3 \) on \( Q_1 \), \( \mathbf{F}_{13} \).2. **Resolve Forces into Components:

A) Draw the free-body diagram of the system.

2. A third charge, Q3, is added to the other two: Q3 F1 3 m Q2 = +3 μC Q1 = +6 μC 4 m A. The electric force on Q1 is now acting to the upper left, at a 45° to the vertical (as shown above). Calculate the unknown charge Q3. B. Using your answer in part (A), determine the electric force acting on Q2. You may answer in vector component form (i.e. F2 = Fx, Fy) 45°

2. A third charge, Q3, is added to the other two: Q3 F1 3 m Q2 = +3 μC Q1 = +6 μC 4 m A. The electric force on Q1 is now acting to the upper left, at a 45° to the vertical (as shown above). Calculate the unknown charge Q3. B. Using your answer in part (A), determine the electric force acting on Q2. You may answer in vector component form (i.e. F2 = Fx, Fy) 45°

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ISBN:9781305952300

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Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

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Question

### Application of Coulomb's Law with a Third Charge

When a third charge, \( Q_3 \), is introduced into the system, the following configuration arises:

#### Diagram Description
- **Charges and Positions:**
- \( Q_1 \) = +6 μC at the origin.
- \( Q_2 \) = +3 μC located 4 meters to the right of \( Q_1 \) on the x-axis.
- \( Q_3 \) is located 3 meters above \( Q_1 \) on the y-axis.
- **Force Vector and Angle:**
- The electric force \( F_1 \) on \( Q_1 \) is acting towards the upper left, at a 45° angle to the vertical.

#### Problem Statement
A. The electric force on \( Q_1 \) is now acting to the upper left, at a 45° angle to the vertical (as shown in the diagram). Calculate the unknown charge \( Q_3 \).

B. Using your answer in part (A), determine the electric force acting on \( Q_2 \). You may answer in vector component form (i.e., \( \mathbf{F_2} = (F_{x}, F_{y}) \)).

### Detailed Analysis

**Diagram Explanation:**
- **\( Q_1 \)** is at the origin.
- **\( Q_2 \)** is 4 meters along the x-axis from \( Q_1 \).
- **\( Q_3 \)** is 3 meters along the y-axis from \( Q_1 \).
- The force \( F_1 \) affects \( Q_1 \) at a 45° angle to the vertical axis heading towards the upper left.

The problem involves using Coulomb’s Law and the principles of vector addition to find the unknown charge \( Q_3 \) based on the described electric forces, and then determining the resulting force on \( Q_2 \).

### Step-by-Step Solution Approach

**Part A:**
1. **Identify Individual Forces Contribution:**
- Calculate the force due to \( Q_2 \) on \( Q_1 \), \( \mathbf{F}_{12} \).
- Calculate the force due to \( Q_3 \) on \( Q_1 \), \( \mathbf{F}_{13} \).

2. **Resolve Forces into Components:

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