origin. What is the total force (magnitude and direction) exerted by these two charges on a negative point charge $$ q 3= $$ -3.00 nC located at $$ x=4.00 \mathrm{~m} $$ ?

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It looks like you're referring to a scenario involving three point charges, with two of them exerting forces on a third charge $$ q_3 = -3.00 \, \text{nC} $$ located at $$ x = 4.00 \, \text{m} $$. To determine the **total force** (both magnitude and direction) exerted on $$ q_3 $$ by the other two charges, we'll need additional information about the other two charges, specifically:

1. **Magnitudes of the Other Charges ($$ q_1 $$ and $$ q_2 $$):**
   - What are the values of $$ q_1 $$ and $$ q_2 $$? Are they positive, negative, or a mix of both?

2. **Positions of the Other Charges:**
   - Where are $$ q_1 $$ and $$ q_2 $$ located relative to $$ q_3 $$? Are they on the same line (e.g., the x-axis), or are they positioned elsewhere in space?

3. **Configuration Details:**
   - Is this a one-dimensional problem (all charges along a straight line) or a two/three-dimensional problem?
   - Are there any symmetries or specific arrangements that can simplify the calculations?

### **General Approach to Solving the Problem:**

Once you have the necessary information, you can follow these steps to determine the total force on $$ q_3 $$:

1. **Identify Each Force:**
   - Use Coulomb's Law to calculate the force exerted by each charge on $$ q_3 $$:
     $$
     F = k_e \frac{|q_i \cdot q_3|}{r^2}
     $$
     where:
     - $$ F $$ is the magnitude of the force,
     - $$ k_e $$ is Coulomb's constant ($$ 8.988 \times 10^9 \, \text{N·m}^2/\text{C}^2 $$),
     - $$ q_i $$ is the other charge,
     - $$ r $$ is the distance between the charges.

2. **Determine the Direction of Each Force:**
   - If the charges are of opposite signs, the force is **attractive** (toward the other charge).
   - If the charges are of the same sign, the force is **repulsive** (away from the other charge).

3. **Break Down Forces into Components (if necessary):**
   - If the charges are not aligned along a single axis, resolve each force into its x and y components.

4. **Sum the Forces Vectorially:**
   - Add the individual force vectors to obtain the **total force** on $$ q_3 $$.

5. **Calculate the Magnitude and Direction:**
   - Use the Pythagorean theorem to find the total magnitude.
   - Use trigonometry (e.g., arctangent) to determine the direction relative to a reference axis.

### **Example Scenario:**

Let's assume a simple case where:
- $$ q_1 = +5.00 \, \text{nC} $$ is located at the origin ($$ x = 0 \, \text{m} $$).
- $$ q_2 = -2.00 \, \text{nC} $$ is located at $$ x = 2.00 \, \text{m} $$.
- $$ q_3 = -3.00 \, \text{nC} $$ is located at $$ x = 4.00 \, \text{m} $$.

**Steps:**

1. **Force Due to $$ q_1 $$ on $$ q_3 $$:**
   - Distance: $$ r_{13} = 4.00 \, \text{m} $$
   - $$ F_{13} = k_e \frac{|q_1 \cdot q_3|}{r_{13}^2} = 8.988 \times 10^9 \cdot \frac{5 \times 10^{-9} \times 3 \times 10^{-9}}{16} \approx 8.988 \times 10^9 \cdot \frac{15 \times 10^{-18}}{16} \approx 8.988 \times 10^9 \cdot 0.9375 \times 10^{-18} \approx 8.438 \times 10^{-9} \, \text{N} $$
   - Direction: Since $$ q_1 $$ is positive and $$ q_3 $$ is negative, the force is **attractive**, directing toward $$ q_1 $$ (to the left).

2. **Force Due to $$ q_2 $$ on $$ q_3 $$:**
   - Distance: $$ r_{23} = 4.00 - 2.00 = 2.00 \, \text{m} $$
   - $$ F_{23} = k_e \frac{|q_2 \cdot q_3|}{r_{23}^2} = 8.988 \times 10^9 \cdot \frac{2 \times 10^{-9} \times 3 \times 10^{-9}}{4} \approx 8.988 \times 10^9 \cdot \frac{6 \times 10^{-18}}{4} \approx 8.988 \times 10^9 \cdot 1.5 \times 10^{-18} \approx 13.482 \times 10^{-9} \, \text{N} $$
   - Direction: $$ q_2 $$ is negative and $$ q_3 $$ is also negative, so the force is **repulsive**, directing away from $$ q_2 $$ (to the right).

3. **Total Force on $$ q_3 $$:**
   - $$ F_{13} $$ acts to the left (-x direction): $$ -8.438 \times 10^{-9} \, \text{N} $$
   - $$ F_{23} $$ acts to the right (+x direction): $$ +13.482 \times 10^{-9} \, \text{N} $$
   - **Net Force:**
     $$
     F_{\text{total}} = F_{23} + F_{13} = 13.482 \times 10^{-9} - 8.438 \times 10^{-9} = 5.044 \times 10^{-9} \, \text{N} \, \text{(to the right)}
     $$
   - **Magnitude:** $$ 5.044 \times 10^{-9} \, \text{N} $$
   - **Direction:** To the right along the x-axis.

### **Conclusion:**

To provide a precise answer to your question, please supply the following details about the two charges exerting forces on $$ q_3 $$:

- **Magnitudes of $$ q_1 $$ and $$ q_2 $$**
- **Positions of $$ q_1 $$ and $$ q_2 $$ relative to $$ q_3 $$**
- **Any additional configuration details (e.g., alignment along axes, angles between forces)**

With this information, I can help you calculate the exact magnitude and direction of the total force exerted on the negative point charge $$ q_3 $$.
To determine the total force on the negative point charge $$ q_3 = -3.00 \, \text{nC} $$ located at $$ x = 4.00 \, \text{m} $$, you need to know the magnitudes and positions of the other two charges ($$ q_1 $$ and $$ q_2 $$) and their arrangement relative to $$ q_3 $$. Once you provide this information, I can calculate the exact magnitude and direction of the total force exerted on $$ q_3 $$.

origin. What is the total force (magnitude and direction) exerted by these two charges on a negative point charge \( q 3= \) -3.00 nC located at \( x=4.00 \mathrm{~m} \) ?

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