Calculate the magnitude of electric force between a proton and an electron that are 83.6 pm apart.

To calculate the magnitude of the electric force between a proton and an electron separated by a distance of \( 83.6 \) pm (picometers), we can use Coulomb's law. Coulomb's law is given by:

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

Where:
- \( F \) is the magnitude of the force between the charges,
- \( k_e \) is Coulomb's constant, \( k_e \approx 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 \cdot \text{C}^{-2} \),
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges, where \( q_1 = q_2 = e \approx 1.6 \times 10^{-19} \, \text{C} \),
- \( r \) is the distance between the charges, which is \( 83.6 \, \text{pm} = 83.6 \times 10^{-12} \, \text{m} \).

Let's plug in the values:

\[
F = (8.99 \times 10^9) \cdot \frac{(1.6 \times 10^{-19})^2}{(83.6 \times 10^{-12})^2}
\]

Now, let's calculate it step by step.

1. Calculate \( (1.6 \times 10^{-19})^2 \):
\[
(1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \, \text{C}^2
\]

2. Calculate \( (83.6 \times 10^{-12})^2 \):
\[
(83.6 \times 10^{-12})^2 = 6.98896 \times 10^{-21} \, \text{m}^2
\]

3. Calculate the force \( F \):
\[
F = (8.99 \times 10^9) \cdot \frac{2.56 \times 10^{-38}}{6.98896 \times 10^{-21}} 
\]

Finally,

\[
F = 3.29 \times 10^{-10} \, \text{N}
\]

So, the magnitude of the electric force between the proton and the electron is approximately \( 3.29 \times 10^{-10} \, \text{N} \).

Would you like more details, or do you have any questions? Here are some related questions:

1. How does the force change if the distance between the charges is doubled?
2. What would be the force if the charges were both positive?
3. How does this force compare with the gravitational force between the same particles?
4. What is the significance of the direction of the electric force?
5. How would the force change if the medium between the charges was not a vacuum?

**Tip:** Remember, the electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Discover how to calculate the magnitude of the electric force between a proton and an electron separated by 83.6 pm using Coulomb's Law. This problem involves key concepts in electrostatics, including the use of Coulomb's constant and the inverse-square law. Suitable for students in Grades 10-12.

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