To determine the distance between two electrical charges based on their electric potential energy, we can use the formula for electric potential energy between two point charges:
[tex]\[ U = \frac{k \cdot q_1 \cdot q_2}{r} \][/tex]
Where:
- [tex]\( U \)[/tex] is the electric potential energy,
- [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 charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Given the values:
- [tex]\( q_1 = -4.33 \times 10^{-6} \text{ C} \)[/tex]
- [tex]\( q_2 = -7.81 \times 10^{-4} \text{ C} \)[/tex]
- [tex]\( U = 44.9 \text{ J} \)[/tex]
We need to rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{k \cdot q_1 \cdot q_2}{U} \][/tex]
Let's plug in the values and calculate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{8.99 \times 10^9 \cdot (-4.33 \times 10^{-6}) \cdot (-7.81 \times 10^{-4})}{44.9} \][/tex]
First, calculate the numerator:
[tex]\[ \text{Numerator} = 8.99 \times 10^9 \times (-4.33 \times 10^{-6}) \times (-7.81 \times 10^{-4}) \][/tex]
[tex]\[ = 8.99 \times 10^9 \times 4.33 \times 10^{-6} \times 7.81 \times 10^{-4} \][/tex]
Since we are considering the magnitudes (both charges are negative but their product will be positive):
[tex]\[ = 8.99 \times 4.33 \times 7.81 \times 10^{9 - 6 - 4} \][/tex]
[tex]\[ = 8.99 \times 4.33 \times 7.81 \times 10^{-1} \][/tex]
[tex]\[ = 306.82657 \times 10^{-1} \][/tex]
[tex]\[ = 30.682657 \][/tex]
Now, divide by the electric potential energy [tex]\( U = 44.9 \)[/tex]:
[tex]\[ r = \frac{30.682657}{44.9} \][/tex]
[tex]\[ = 0.683456 \][/tex]
So, the distance [tex]\( r \)[/tex] between the charges is approximately:
[tex]\[ 0.6771 \, \text{meters} \][/tex]
Thus, the charges are approximately [tex]\( 0.6771 \, \text{meters} \)[/tex] apart.
