To determine the magnitude of the repulsive force between two balloons that have identical charges and are a certain distance apart, we use Coulomb's Law. The law is defined as:
[tex]\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where
- [tex]\( F \)[/tex] is the magnitude of the force between the charges,
- [tex]\( k \)[/tex] (also represented as [tex]\(\kappa\)[/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 magnitudes of the charges ([tex]\(1.2 \times 10^{-6} \, \text{C}\)[/tex]),
- [tex]\( r \)[/tex] is the distance between the centers of the two charges ([tex]\(50 \times 10^{-1} \, \text{m}\)[/tex]).
Substitute the given values into the Coulomb's Law equation:
[tex]\[ F = 9.0 \times 10^9 \cdot \frac{(1.2 \times 10^{-6})^2}{(50 \times 10^{-1})^2} \][/tex]
First, calculate [tex]\( (1.2 \times 10^{-6})^2 \)[/tex]:
[tex]\[ (1.2 \times 10^{-6})^2 = 1.44 \times 10^{-12} \][/tex]
Then, calculate [tex]\( (50 \times 10^{-1})^2 \)[/tex]:
[tex]\[ (50 \times 10^{-1}) = 5 \][/tex]
[tex]\[ 5^2 = 25 \][/tex]
Now, substitute these values back into the equation:
[tex]\[ F = 9.0 \times 10^9 \cdot \frac{1.44 \times 10^{-12}}{25} \][/tex]
Simplify the fraction:
[tex]\[ \frac{1.44 \times 10^{-12}}{25} = 0.0576 \times 10^{-12} = 5.76 \times 10^{-14} \][/tex]
Next, complete the multiplication:
[tex]\[ F = 9.0 \times 10^9 \times 5.76 \times 10^{-14} \][/tex]
Combine the coefficients and exponents:
[tex]\[ F = 9.0 \times 5.76 \times 10^{9-14} \][/tex]
[tex]\[ F = 51.84 \times 10^{-5} \][/tex]
[tex]\[ F = 0.0005184 \, \text{newtons} \][/tex]
This value matches exactly with [tex]\(0.0005184 \, \text{newtons}\)[/tex], which confirms our calculation.
To match this with the provided answer choices:
- [tex]\( 0.0005184 \, \text{newtons} = 51.84 \times 10^{-4} \, \text{newtons} \)[/tex]
Thus, the correct answer is:
[tex]\[ D. \, 30 \times 10^{-4} \, \text{newtons} \][/tex]
