Three identical metal spheres are hung from a ceiling on rigid non-conducting rods. Sphere 1, sphere 2, and sphere 3 have charges
31 µC,−50 µC, and 46 µC, respectively. Each sphere is separated by
11 cm.  What is the net electric force on the middle sphere due to spheres 1 and 3? (Enter the magnitude only.)

To find the net electric force on the middle sphere (Sphere 2) due to Spheres 1 and 3, we need to use Coulomb's law. The force between two charged objects is given by:

\[
F = k_e \frac{|q_1 \cdot q_2|}{r^2}
\]

Where:
- \( F \) is the magnitude of the force between the charges.
- \( k_e \) is Coulomb's constant, \( k_e = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
- \( q_1 \) and \( q_2 \) are the charges on the two spheres.
- \( r \) is the distance between the two charges.

### Step 1: Calculate the force between Sphere 1 and Sphere 2

Given:
- \( q_1 = 31 \, \mu C = 31 \times 10^{-6} \, C \)
- \( q_2 = -50 \, \mu C = -50 \times 10^{-6} \, C \)
- \( r_{12} = 11 \, cm = 0.11 \, m \)

The force \( F_{12} \) between Sphere 1 and Sphere 2 is:

\[
F_{12} = k_e \frac{|q_1 \cdot q_2|}{r_{12}^2}
\]

### Step 2: Calculate the force between Sphere 2 and Sphere 3

Given:
- \( q_2 = -50 \, \mu C = -50 \times 10^{-6} \, C \)
- \( q_3 = 46 \, \mu C = 46 \times 10^{-6} \, C \)
- \( r_{23} = 11 \, cm = 0.11 \, m \)

The force \( F_{23} \) between Sphere 2 and Sphere 3 is:

\[
F_{23} = k_e \frac{|q_2 \cdot q_3|}{r_{23}^2}
\]

### Step 3: Determine the direction and net force

- The force \( F_{12} \) will be attractive (towards Sphere 1) because Sphere 1 is positively charged and Sphere 2 is negatively charged.
- The force \( F_{23} \) will also be attractive (towards Sphere 3) because Sphere 3 is positively charged and Sphere 2 is negatively charged.

Since both forces are in opposite directions along the same line, the net force \( F_{net} \) on Sphere 2 is the difference between \( F_{23} \) and \( F_{12} \).

Let's calculate the magnitudes of these forces.

I'll perform the calculations now.The net electric force on the middle sphere (Sphere 2) due to Spheres 1 and 3 is approximately **557.23 N**.

Would you like further details or have any questions about the calculations? Here are some related questions to consider:

1. How would the net force change if the distance between the spheres was doubled?
2. What is the direction of the net force on Sphere 2?
3. How would the net force on Sphere 2 be affected if the charges on Spheres 1 and 3 were reversed?
4. What is the significance of the sign of the charges in determining the direction of the force?
5. How does Coulomb’s law relate to Newton's law of gravitation?

**Tip:** Remember that the direction of the electric force depends on whether the interacting charges are like or unlike. Like charges repel, while unlike charges attract.

Learn how to calculate the net electric force on the middle sphere due to charges using Coulomb's law. Given three metal spheres with specific charges and separations, find the magnitude of the force exerted on the middle sphere. Detailed step-by-step solution provided.

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