To determine which pair of charges result in the largest repulsive force, we'll utilize the Coulomb's Law equation:
[tex]\[ F = \frac{k \cdot q_1 \cdot q_2}{R^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the force between two charges,
- [tex]\( k \)[/tex] is Coulomb's constant,
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the two charges,
- [tex]\( R \)[/tex] is the distance between the charges.
Given pairs of charges are:
1. [tex]\(-3q\)[/tex] and [tex]\(-2q\)[/tex]
2. [tex]\(+3q\)[/tex] and [tex]\(-2q\)[/tex]
3. [tex]\(-2q\)[/tex] and [tex]\(+4q\)[/tex]
We should calculate the force for each given pair to determine which one has the largest repulsive force.
### Step-by-step Calculation:
#### Pair 1: [tex]\(-3q\)[/tex] and [tex]\(-2q\)[/tex]
[tex]\[ F_1 = \frac{k \cdot (-3q) \cdot (-2q)}{R^2} = \frac{k \cdot 6q^2}{R^2} \][/tex]
Since both charges are negative, the product [tex]\((-3q) \cdot (-2q)\)[/tex] is positive, leading to a repulsive force.
#### Pair 2: [tex]\(+3q\)[/tex] and [tex]\(-2q\)[/tex]
[tex]\[ F_2 = \frac{k \cdot (+3q) \cdot (-2q)}{R^2} = \frac{k \cdot (-6q^2)}{R^2} \][/tex]
The product [tex]\((+3q) \cdot (-2q)\)[/tex] is negative, leading to an attractive force, not repulsive.
#### Pair 3: [tex]\(-2q\)[/tex] and [tex]\(+4q\)[/tex]
[tex]\[ F_3 = \frac{k \cdot (-2q) \cdot (+4q)}{R^2} = \frac{k \cdot (-8q^2)}{R^2} \][/tex]
The product [tex]\((-2q) \cdot (+4q)\)[/tex] is negative, leading to an attractive force, not repulsive.
### Comparing Repulsive Forces:
Only Pair 1 results in a repulsive force. Therefore, the force corresponding to Pair 1 is:
[tex]\[ F_1 = \frac{6kq^2}{R^2} \][/tex]
After following through with the calculations, we see that the:
- Force for Pair 1 is [tex]\( 6.0 \)[/tex] units (repulsive force),
- Forces for Pair 2 and Pair 3 are -6.0 and -8.0 units (attractive forces).
Therefore, the largest repulsive force is produced by the pair of charges:
[tex]\(\boxed{-3q \text{ and } -2q}\)[/tex]
