Sure! To determine the value of the second charge [tex]\( q_2 \)[/tex] given the information in the problem, we can use Coulomb's Law. Coulomb's Law states that the force between two charges is given by:
[tex]\[ F = k \frac{|q_1 q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the force between the charges,
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges,
- [tex]\( k \)[/tex] is Coulomb’s constant, approximately [tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 \cdot \text{C}^{-2} \)[/tex].
Given:
- [tex]\( q_1 = -0.00325 \, \text{C} \)[/tex]
- [tex]\( F = 48900 \, \text{N} \)[/tex]
- [tex]\( r = 5.62 \, \text{m} \)[/tex]
We need to solve for [tex]\( q_2 \)[/tex]. First, we rearrange Coulomb's Law to solve for [tex]\( q_2 \)[/tex]:
[tex]\[ q_2 = \frac{F \cdot r^2}{k \cdot |q_1|} \][/tex]
Now, substitute the given values into the formula:
[tex]\[ q_2 = \frac{48900 \cdot (5.62)^2}{8.99 \times 10^9 \cdot | -0.00325 |} \][/tex]
Evaluating the expression inside gives us:
[tex]\[ q_2 = \frac{48900 \cdot 31.5844}{8.99 \times 10^9 \cdot 0.00325} \][/tex]
Let's simplify this step-by-step:
1. Compute the numerator:
[tex]\[ 48900 \cdot 31.5844 \approx 1544642.76 \][/tex]
2. Compute the denominator:
[tex]\[ 8.99 \times 10^9 \cdot 0.00325 \approx 2.92175 \times 10^7 \][/tex]
3. Divide the numerator by the denominator:
[tex]\[ q_2 = \frac{1544642.76}{2.92175 \times 10^7} \approx 0.05286137280739284 \][/tex]
Since the force is repelling and [tex]\( q_1 \)[/tex] is negative, [tex]\( q_2 \)[/tex] should have the same sign as [tex]\( q_1 \)[/tex]. Therefore, [tex]\( q_2 \)[/tex] is also negative.
The value of the second charge [tex]\( q_2 \)[/tex] is:
[tex]\[ q_2 \approx -0.0529 \, \text{C} \][/tex]
Thus, the charge [tex]\( q_2 \)[/tex] is approximately [tex]\( -0.0529 \, \text{C} \)[/tex].
