To determine the electric force between two charged spheres, we'll use Coulomb's Law. Coulomb's Law provides a way to calculate the electric force between two point charges. The formula is given by:
[tex]\[ F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the electric force between the charges,
- [tex]\( k \)[/tex] is Coulomb's constant ([tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex]),
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the two charges,
- [tex]\( r \)[/tex] is the separation distance between the centers of the two charges.
Given:
- [tex]\( q_1 = -3.0 \times 10^7 \, \text{C} \)[/tex]
- [tex]\( q_2 = -3.0 \times 10^7 \, \text{C} \)[/tex]
- [tex]\( r = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)[/tex]
Since the charges are the same, we can simplify the absolute value term to just the magnitude:
[tex]\[ |q_1 \cdot q_2| = |-3.0 \times 10^7 \, \text{C} \times -3.0 \times 10^7 \, \text{C}| = 9.0 \times 10^{14} \, \text{C}^2 \][/tex]
Next, we substitute these values into Coulomb's Law equation:
[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2 / \text{C}^2 \times \frac{9.0 \times 10^{14} \, \text{C}^2}{(2 \times 10^{-3} \, \text{m})^2} \][/tex]
Now, calculating the denominator which is [tex]\( r^2 \)[/tex]:
[tex]\[ (2 \times 10^{-3} \, \text{m})^2 = 4 \times 10^{-6} \, \text{m}^2 \][/tex]
So, our equation becomes:
[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2 / \text{C}^2 \times \frac{9.0 \times 10^{14} \, \text{C}^2}{4 \times 10^{-6} \, \text{m}^2} \][/tex]
Simplifying the fraction:
[tex]\[ \frac{9.0 \times 10^{14} \, \text{C}^2}{4 \times 10^{-6} \, \text{m}^2} = 2.25 \times 10^{20} \, \text{C}^2 / \text{m}^2 \][/tex]
Then, multiply this by Coulomb's constant:
[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2 / \text{C}^2 \times 2.25 \times 10^{20} \, \text{C}^2 / \text{m}^2 \][/tex]
[tex]\[ F = 2.02275 \times 10^{30} \, \text{N} \][/tex]
Thus, the electric force between the two charged spheres is [tex]\( 2.02275 \times 10^{30} \, \text{N} \)[/tex].
