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A charge of [tex]-0.00325 \, \text{C}[/tex] ([tex]\(q_1\)[/tex]) is placed 5.62 m from a second charge ([tex]\(q_2\)[/tex]). The first charge is repelled with a force of 48,900 N. What is the value of [tex]\(q_2\)[/tex]? Include the sign of the charge (+ or -).

[tex]\[ ? \, \text{C} \][/tex]



Answer :

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]

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