To find the force of interaction between the two charged balloons, we use Coulomb's Law. The force [tex]\( F \)[/tex] between two point charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
Where:
- [tex]\( k \)[/tex] is Coulomb's constant [tex]\( (9.0 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2) \)[/tex]
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the charges (both [tex]\(+2.30 \times 10^{-10} \, \text{C}) \)[/tex]
- [tex]\( r \)[/tex] is the separation distance between the charges ( [tex]\(3.20 \times 10^{-1} \, \text{m}) \)[/tex]
Let's input the given values into the formula step by step:
1. Identify the charges:
[tex]\[ q_1 = 2.30 \times 10^{-10} \, \text{C} \][/tex]
[tex]\[ q_2 = 2.30 \times 10^{-10} \, \text{C} \][/tex]
2. Identify the separation distance:
[tex]\[ r = 3.20 \times 10^{-1} \, \text{m} \][/tex]
3. Identify Coulomb's constant:
[tex]\[ k = 9.0 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \][/tex]
4. Plug in these values into Coulomb's law:
[tex]\[ F = 9.0 \times 10^9 \times \frac{(2.30 \times 10^{-10}) \times (2.30 \times 10^{-10})}{(3.20 \times 10^{-1})^2} \][/tex]
Performing the calculations:
- Calculate the product of the charges:
[tex]\[ 2.30 \times 10^{-10} \times 2.30 \times 10^{-10} = 5.29 \times 10^{-20} \][/tex]
- Calculate the square of the distance:
[tex]\[ (3.20 \times 10^{-1})^2 = 1.024 \times 10^{-1} \][/tex]
- Compute the force:
[tex]\[ F = 9.0 \times 10^9 \times \frac{5.29 \times 10^{-20}}{1.024 \times 10^{-1}} \][/tex]
Thus:
[tex]\[ F \approx 9.0 \times 10^9 \times 5.168 \times 10^{-19} = 4.6494140625 \times 10^{-9} \, \text{N} \][/tex]
So, the force of interaction between the two balloons is approximately:
[tex]\[ \boxed{4.6494140625 \times 10^{-9} \, \text{N}} \][/tex]
