To determine the distance [tex]\( r \)[/tex] between two point charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] such that the electrostatic force between them equals 8.28 N, we use Coulomb's law. Coulomb's law states that the magnitude of the electrostatic force [tex]\( F \)[/tex] between two point charges is given by:
[tex]\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where,
- [tex]\( F \)[/tex] is the force between the charges (8.28 N),
- [tex]\( k \)[/tex] is Coulomb's constant ([tex]\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)[/tex]),
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
1. Convert the charges to Coulombs:
[tex]\[
q_1 = 20.8 \, \mu\text{C} = 20.8 \times 10^{-6} \, \text{C}
\][/tex]
[tex]\[
q_2 = -69.3 \, \mu\text{C} = -69.3 \times 10^{-6} \, \text{C}
\][/tex]
2. Determine the absolute value of the product of the charges:
[tex]\[
|q_1 \cdot q_2| = |(20.8 \times 10^{-6}) \cdot (-69.3 \times 10^{-6})|
\][/tex]
[tex]\[
|q_1 \cdot q_2| = 1.44144 \times 10^{-9} \, \text{C}^2
\][/tex]
3. Calculate the numerator [tex]\( k \cdot |q_1 \cdot q_2| \)[/tex]:
[tex]\[
k \cdot |q_1 \cdot q_2| = (8.99 \times 10^9) \cdot (1.44144 \times 10^{-9})
\][/tex]
[tex]\[
k \cdot |q_1 \cdot q_2| = 12.9585456 \, \text{N·m}^2
\][/tex]
4. Solve for the distance [tex]\( r \)[/tex] using the rearranged formula:
[tex]\[
r = \sqrt{\frac{k \cdot |q_1 \cdot q_2|}{F}}
\][/tex]
[tex]\[
r = \sqrt{\frac{12.9585456}{8.28}}
\][/tex]
[tex]\[
r = \sqrt{1.565592}
\][/tex]
[tex]\[
r = 1.251 \, \text{m}
\][/tex]
Therefore, the distance between the charges must be [tex]\( \boxed{1.251} \)[/tex] meters for the electrostatic force between them to have a magnitude of 8.28 N.
