Certainly! Let's find the force between two charges [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex] using Coulomb's Law. The given values are:
- Coulomb's constant, [tex]\( k_e = 8.99 \times 10^9 \, \text{Nm}^2 / \text{C}^2 \)[/tex]
- Charges, [tex]\( q_1 = 1 \, \text{C} \)[/tex] and [tex]\( q_3 = 1 \, \text{C} \)[/tex]
- Distance between the charges, [tex]\( r = 0.55 \, \text{m} \)[/tex]
Coulomb's Law formula for the force between two point charges is given by:
[tex]\[ \vec{F}_3 = k_e \frac{|q_1 q_3|}{r^2} \][/tex]
Now, let's substitute the given values into the formula:
1. Check the magnitudes of the charges:
[tex]\[ |q_1 q_3| = |1 \cdot 1| = 1 \, \text{C}^2 \][/tex]
2. Substitute [tex]\( k_e = 8.99 \times 10^9 \, \text{Nm}^2 / \text{C}^2 \)[/tex], [tex]\( |q_1 q_3| = 1 \, \text{C}^2 \)[/tex], and [tex]\( r = 0.55 \, \text{m} \)[/tex] into the formula:
[tex]\[ \vec{F}_3 = 8.99 \times 10^9 \cdot \frac{1}{(0.55)^2} \][/tex]
3. Calculate the square of the distance:
[tex]\[ (0.55)^2 = 0.3025 \, \text{m}^2 \][/tex]
4. Now, divide the numerator by the square of the distance:
[tex]\[ \frac{1}{0.3025} \approx 3.304 \][/tex]
5. Finally, multiply this result by [tex]\( 8.99 \times 10^9 \)[/tex]:
[tex]\[ \vec{F}_3 \approx 8.99 \times 10^9 \cdot 3.304 \][/tex]
6. Perform the multiplication:
[tex]\[ \vec{F}_3 \approx 29.719 \times 10^9 \, \text{N} \][/tex]
[tex]\[ \vec{F}_3 \approx 2.9719 \times 10^{10} \, \text{N} \][/tex]
Therefore, the force [tex]\( \vec{F}_3 \)[/tex] between the charges [tex]\( q_1 \)[/tex] and [tex]\( q_3 \)[/tex] is approximately [tex]\( 2.9719 \times 10^{10} \, \text{N} \)[/tex].
