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This image depicts a physics problem involving the electric force between three point charges arranged in a 2D plane. The task is to find the magnitude and direction of the electric force on the charge \( q = 5.10 \, \text{nC} \) at the origin due to two other charges. Here's a breakdown of how to approach the problem:

### Step 1: Use Coulomb's Law
The electric force \( \mathbf{F} \) between two point charges is given by Coulomb’s Law:
\[
\mathbf{F} = k_e \frac{|q_1 q_2|}{r^2} \hat{r}
\]
where:
- \( k_e = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \) is the Coulomb constant,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges,
- \( \hat{r} \) is the unit vector in the direction from one charge to the other.

### Step 2: Calculate the Forces from Each Charge
- **Force due to the \( 6.00 \, \text{nC} \) charge** located at \( r_{12} = 0.265 \, \text{m} \) to the right of \( q \):
  \[
  F_1 = k_e \frac{|(5.10 \times 10^{-9}) (6.00 \times 10^{-9})|}{(0.265)^2}
  \]

- **Force due to the \( -3.00 \, \text{nC} \) charge** located directly below \( q \) at \( 0.100 \, \text{m} \):
  \[
  F_2 = k_e \frac{|(5.10 \times 10^{-9}) (-3.00 \times 10^{-9})|}{(0.100)^2}
  \]

### Step 3: Resolve the Forces into Components
Since the forces are vector quantities, we need to resolve them into x and y components:
- The force due to the \( 6.00 \, \text{nC} \) charge acts entirely in the horizontal (+x) direction.
- The force due to the \( -3.00 \, \text{nC} \) charge acts entirely in the vertical (-y) direction.

### Step 4: Find the Resultant Force
Once the x and y components of the individual forces are known, combine them to find the total force in both directions. The magnitude of the resultant force is given by:
\[
F_{\text{total}} = \sqrt{F_x^2 + F_y^2}
\]
The direction \( \theta \) of the force can be found using:
\[
\theta = \tan^{-1} \left(\frac{F_y}{F_x}\right)
\]
This angle is measured counterclockwise from the +x-axis.

Do you want the detailed computation of these steps? Here are 5 related questions to explore further:

1. How do we calculate the electric force for a charge configuration in 3D?
2. What happens to the electric force if the charges are increased by a factor of 2?
3. How does the direction of the force change if the charges are moved?
4. What are the effects of dielectric materials on electric forces between charges?
5. How can we extend this analysis to a continuous charge distribution?

**Tip**: When working with vector forces, always resolve forces into components before combining them to find the resultant force.

Three point charges are arranged as shown in the figure. Find the magnitude and direction of the electric force on the particle q = 5.10 nC at the origin.

This problem involves calculating the electric force on a charge q = 5.10 nC at the origin due to two other charges using Coulomb's law. The problem uses key electrostatic concepts, vector addition, and trigonometry. The solution provides step-by-step calculation for force magnitude and direction.

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