Two uncharged spheres are separated by 2.80 m. If 4.10 ✕ 1012 electrons are removed from one sphere and placed on the other, determine the magnitude of the Coulomb force (in N) on one of the spheres, treating the spheres as point charges.

To determine the magnitude of the Coulomb force between two spheres that are separated by a distance and have unequal charges, we will use Coulomb's Law:

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

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

### Step 1: Determine the charge on each sphere
When electrons are removed from one sphere and placed on the other, one sphere gains a negative charge, and the other sphere becomes positively charged by the same amount. The charge due to removing \( 4.10 \times 10^{12} \) electrons can be calculated using the elementary charge of an electron \( e = 1.602 \times 10^{-19} \, \text{C} \).

\[
q = N \cdot e
\]
Where \( N = 4.10 \times 10^{12} \) is the number of electrons.

\[
q = (4.10 \times 10^{12}) \cdot (1.602 \times 10^{-19}) = 6.5682 \times 10^{-7} \, \text{C}
\]

Thus, the charge on each sphere is:
- \( q_1 = 6.5682 \times 10^{-7} \, \text{C} \) (positive charge),
- \( q_2 = -6.5682 \times 10^{-7} \, \text{C} \) (negative charge).

### Step 2: Apply Coulomb's Law
The distance between the spheres is given as \( r = 2.80 \, \text{m} \). Now, we can plug these values into Coulomb's Law:

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

Since \( q_1 \) and \( q_2 \) are equal in magnitude but opposite in sign, the product \( q_1 \cdot q_2 \) is:

\[
q_1 \cdot q_2 = (6.5682 \times 10^{-7}) \cdot (-6.5682 \times 10^{-7}) = -4.3154 \times 10^{-13} \, \text{C}^2
\]

The magnitude of the product is \( |q_1 \cdot q_2| = 4.3154 \times 10^{-13} \, \text{C}^2 \).

Now calculate the force:

\[
F = (8.99 \times 10^9) \cdot \frac{4.3154 \times 10^{-13}}{(2.80)^2}
\]

\[
F = (8.99 \times 10^9) \cdot \frac{4.3154 \times 10^{-13}}{7.84}
\]

\[
F = (8.99 \times 10^9) \cdot 5.5056 \times 10^{-14}
\]

\[
F = 4.95 \times 10^{-4} \, \text{N}
\]

Thus, the magnitude of the Coulomb force is approximately:

\[
F \approx 4.95 \times 10^{-4} \, \text{N}
\]

### Summary:
The magnitude of the Coulomb force between the two spheres is \( 4.95 \times 10^{-4} \, \text{N} \).

Would you like further details or have any questions? Here are five related questions to expand this topic:

1. What would happen to the force if the separation between the spheres was halved?
2. How does the force change if instead of removing electrons, protons are added?
3. Can you explain how the direction of the force would differ between attraction and repulsion?
4. What is the significance of treating the spheres as point charges in this context?
5. How would the Coulomb force change if the number of electrons removed was doubled?

**Tip:** Remember, the Coulomb force is inversely proportional to the square of the distance between the charges, so even small changes in distance can have a significant impact on the force!

Two uncharged spheres are separated by 2.80 m. If 4.10 ✕ 10^12 electrons are removed from one sphere and placed on the other, determine the magnitude of the Coulomb force (in N) on one of the spheres, treating the spheres as point charges.

This physics problem involves calculating the Coulomb force between two spheres separated by 2.80 m after transferring 4.10 ✕ 10^12 electrons. Learn how to apply Coulomb's Law and use the elementary charge of an electron for detailed calculation. Suitable for students in Grades 10-12.

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