Let's solve the problem step-by-step using Coulomb's law.
### Step-by-Step Solution:
1. Given Data:
- Charge of particle 1, [tex]\( q_1 = -6.25 \times 10^{-9} \)[/tex] C
- Charge of particle 2, [tex]\( q_2 = 2.91 \times 10^{-9} \)[/tex] C
- Initial distance between the particles, [tex]\( r_{\text{initial}} = 0.38 \)[/tex] m
- Coulomb's constant, [tex]\( k = 9.00 \times 10^9 \)[/tex] N·m²/C²
2. Calculate the Initial Force:
Using Coulomb's law, the force [tex]\( F_e \)[/tex] between two charges is given by:
[tex]\[
F_e = \frac{k q_1 q_2}{r^2}
\][/tex]
For the initial distance [tex]\( r_{\text{initial}} = 0.38 \)[/tex] m:
[tex]\[
F_{\text{initial}} = \frac{(9.00 \times 10^9) \times (-6.25 \times 10^{-9}) \times (2.91 \times 10^{-9})}{(0.38)^2}
\][/tex]
3. Initial Force Calculation Result:
The result of this calculation is:
[tex]\[
F_{\text{initial}} \approx -1.13 \times 10^{-6} \, \text{N}
\][/tex]
This indicates that the initial force is approximately [tex]\(-1.13 \times 10^{-6}\)[/tex] N. This is the electrostatic force between the two charges before the distance is changed.
4. New Distance:
The distance between the particles is cut in half:
[tex]\[
r_{\text{new}} = \frac{r_{\text{initial}}}{2} = \frac{0.38}{2} = 0.19 \, \text{m}
\][/tex]
5. Calculate the New Force:
Using Coulomb's law again for the new distance:
[tex]\[
F_{\text{new}} = \frac{(9.00 \times 10^9) \times (-6.25 \times 10^{-9}) \times (2.91 \times 10^{-9})}{(0.19)^2}
\][/tex]
6. New Force Calculation Result:
The result of this calculation is:
[tex]\[
F_{\text{new}} \approx -4.53 \times 10^{-6} \, \text{N}
\][/tex]
This means that when the distance is cut in half, the new force is approximately [tex]\(-4.53 \times 10^{-6}\)[/tex] N.
### Conclusion:
So, given the options:
A. [tex]\(-4.53 \times 10^{-6} \, \text{N}\)[/tex]
B. [tex]\(-1.13 \times 10^{-6} \, \text{N}\)[/tex]
C. [tex]\(1.13 \times 10^{-6} \, \text{N}\)[/tex]
D. [tex]\(4.53 \times 10^{-6} \, \text{N}\)[/tex]
The initial force between the particles is [tex]\(-1.13 \times 10^{-6} \, \text{N}\)[/tex] and the force when the distance is cut in half is [tex]\(-4.53 \times 10^{-6} \, \text{N}\)[/tex].
Thus, the answer is:
A. [tex]\(-4.53 \times 10^{-6} \, \text{N}\)[/tex]
