To calculate the force between charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] using Coulomb's Law, you can follow these steps:
1. Write down Coulomb’s Law:
[tex]\[
F = k_e \frac{|q_1 q_2|}{r^2}
\][/tex]
Where:
- [tex]\( F \)[/tex] is the magnitude of the force between the charges,
- [tex]\( k_e \)[/tex] is Coulomb’s constant [tex]\( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \)[/tex],
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges [tex]\( (13.0 \times 10^{-6} \, \text{C} \)[/tex] and [tex]\( 7.70 \times 10^{-6} \, \text{C} \)[/tex] respectively),
- [tex]\( r \)[/tex] is the distance between the charges [tex]\( (0.25 \, \text{m}) \)[/tex].
2. Substitute the given values into the formula:
[tex]\[
F = 8.99 \times 10^9 \frac{\text{Nm}^2}{\text{C}^2} \times \frac{|13.0 \times 10^{-6} \, \text{C} \times 7.70 \times 10^{-6} \, \text{C}|}{(0.25 \, \text{m})^2}
\][/tex]
3. Calculate the numerator:
[tex]\[
|q_1 q_2| = |13.0 \times 10^{-6} \times 7.70 \times 10^{-6}| = 1.001 \times 10^{-10} \, \text{C}^2
\][/tex]
4. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = (0.25 \, \text{m})^2 = 0.0625 \, \text{m}^2
\][/tex]
5. Combine the results to find the force:
[tex]\[
F = 8.99 \times 10^9 \frac{\text{Nm}^2}{\text{C}^2} \times \frac{1.001 \times 10^{-10} \, \text{C}^2}{0.0625 \, \text{m}^2}
\][/tex]
6. Simplify the division:
[tex]\[
\frac{1.001 \times 10^{-10} \, \text{C}^2}{0.0625 \, \text{m}^2} = 1.6016 \times 10^{-9} \, \frac{\text{C}^2}{\text{m}^2}
\][/tex]
7. Complete the multiplication:
[tex]\[
F = 8.99 \times 10^9 \times 1.6016 \times 10^{-9} = 14.398384 \, \text{N}
\][/tex]
Therefore, the magnitude of the force between [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] is [tex]\( 14.398383999999998 \, \text{N} \)[/tex].
