To determine the value of [tex]\( q_2 \)[/tex], let's follow the steps below using Coulomb's Law:
1. Understanding Coulomb's Law: It states that the force [tex]\( F \)[/tex] between two point charges is given by the formula:
[tex]\[
F = k \frac{|q_1 \cdot q_2|}{r^2}
\][/tex]
where:
- [tex]\( F \)[/tex] is the magnitude of the force between the charges (in Newtons),
- [tex]\( k \)[/tex] is Coulomb's constant ([tex]\( 8.988 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex]),
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges (in Coulombs),
- [tex]\( r \)[/tex] is the distance between the charges (in meters).
2. Given Values:
- Coulomb's constant [tex]\( k = 8.988 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex],
- Charge [tex]\( q_1 = -0.00325 \, \text{C} \)[/tex],
- Distance [tex]\( r = 5.62 \, \text{m} \)[/tex],
- Force [tex]\( F = 48900 \, \text{N} \)[/tex].
3. Rearranging Coulomb's Law to solve for [tex]\( q_2 \)[/tex]:
[tex]\[
F = k \frac{|q_1 \cdot q_2|}{r^2}
\][/tex]
[tex]\[
|q_2| = \frac{F \cdot r^2}{k \cdot |q_1|}
\][/tex]
4. Plugging in the given values:
[tex]\[
|q_2| = \frac{48900 \cdot (5.62)^2}{8.988 \times 10^9 \cdot | -0.00325|}
\][/tex]
Compute the distance squared:
[tex]\[
(5.62)^2 = 31.5844
\][/tex]
Calculate the numerator:
[tex]\[
48900 \cdot 31.5844 = 1545207.16
\][/tex]
Calculate the denominator:
[tex]\[
8.988 \times 10^9 \cdot 0.00325 = 29206
\][/tex]
Find [tex]\( |q_2| \)[/tex]:
[tex]\[
|q_2| = \frac{1545207.16}{29206} \approx 0.052873135462668176
\][/tex]
5. Determining the sign of [tex]\( q_2 \)[/tex]:
Since the force between the two charges is repulsive, [tex]\( q_2 \)[/tex] must have the same sign as [tex]\( q_1 \)[/tex]. Given that [tex]\( q_1 \)[/tex] is negative, [tex]\( q_2 \)[/tex] must also be negative.
Hence, the value of [tex]\( q_2 \)[/tex] is:
[tex]\[
q_2 \approx -0.052873 \, \text{C}
\][/tex]
