To find the electric force between a positive charge and a negative charge separated by a distance of [tex]\( 10^{-15} \)[/tex] meters using Coulomb's law, we can follow these steps:
1. Identify the Given Values:
- [tex]\( e = 1.6 \times 10^{-19} \, \text{C} \)[/tex] (magnitude of charge of an electron)
- [tex]\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex] (Coulomb's constant)
- [tex]\( r = 10^{-15} \, \text{m} \)[/tex] (distance between charges)
- The charges are [tex]\( q_1 = e \)[/tex] and [tex]\( q_2 = -e \)[/tex] (positive and negative charge)
2. Write Coulomb's Law Formula:
- Coulomb's law states: [tex]\( F = k \frac{|q_1 \cdot q_2|}{r^2} \)[/tex]
3. Substitute the Given Values into the Formula:
- [tex]\( q_1 = 1.6 \times 10^{-19} \, \text{C} \)[/tex]
- [tex]\( q_2 = -1.6 \times 10^{-19} \, \text{C} \)[/tex]
- Absolute value of the product of [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] is [tex]\( |q_1 \cdot q_2| = |(1.6 \times 10^{-19}) \cdot (-1.6 \times 10^{-19})| = 2.56 \times 10^{-38} \, \text{C}^2 \)[/tex]
4. Calculate the Force Magnitude:
- Substitute the values into the formula:
[tex]\[
\text{Force magnitude} = F = k \frac{|q_1 \cdot q_2|}{r^2} = 8.99 \times 10^9 \frac{2.56 \times 10^{-38}}{(10^{-15})^2}
\][/tex]
- Simplify the denominator:
[tex]\[
(10^{-15})^2 = 10^{-30}
\][/tex]
- So the expression becomes:
[tex]\[
F = 8.99 \times 10^9 \times \frac{2.56 \times 10^{-38}}{10^{-30}}
\][/tex]
5. Simplify the Exponents and Calculate the Result:
- Simplify the fraction:
[tex]\[
\frac{2.56 \times 10^{-38}}{10^{-30}} = 2.56 \times 10^{-8}
\][/tex]
- So the expression becomes:
[tex]\[
F = 8.99 \times 10^9 \times 2.56 \times 10^{-8} = 230.144 \, \text{N}
\][/tex]
6. Determine the Nature of the Force:
- Since one charge is positive and one is negative, the force is attractive. This means the force should be directed towards each charge, so the force should be negative.
- Therefore, the force is: [tex]\( F = -230.144 \, \text{N} \)[/tex]
7. Choose the Correct Option:
Given the options:
- 230 N
- [tex]\(-230 \, \text{N}\)[/tex]
- 120 N
- [tex]\(-120 \, \text{N}\)[/tex]
Based on our calculation, the correct choice is [tex]\(-230 \, \text{N}\)[/tex].
