What is the electric force acting between two charges of [tex]0.0042 \, \text{C}[/tex] and [tex]-0.0050 \, \text{C}[/tex] that are [tex]0.0030 \, \text{m}[/tex] apart?

Use [tex]F_e = \frac{k q_1 q_2}{r^2}[/tex] and [tex]k = 9.00 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2[/tex].

A. [tex]-6.3 \times 10^7 \, \text{N}[/tex]
B. [tex]-2.1 \times 10^{10} \, \text{N}[/tex]
C. [tex]6.3 \times 10^7 \, \text{N}[/tex]
D. [tex]2.1 \times 10^{10} \, \text{N}[/tex]



Answer :

To find the electric force between the two charges, we use Coulomb's Law, which is given by the formula:

[tex]\[ F_e = \frac{k q_1 q_2}{r^2} \][/tex]

where:
- [tex]\( F_e \)[/tex] is the electric force between the charges,
- [tex]\( k \)[/tex] is Coulomb's constant, [tex]\( 9.00 \times 10^9 \, N \cdot m^2 / C^2 \)[/tex],
- [tex]\( q_1 \)[/tex] is the magnitude of the first charge, [tex]\( 0.0042 \, C \)[/tex],
- [tex]\( q_2 \)[/tex] is the magnitude of the second charge, [tex]\( -0.0050 \, C \)[/tex],
- [tex]\( r \)[/tex] is the distance between the charges, [tex]\( 0.0030 \, m \)[/tex].

### Step-by-Step Solution:

1. Identify the values in the problem:
[tex]\[ k = 9.00 \times 10^9 \, N \cdot m^2 / C^2 \][/tex]
[tex]\[ q_1 = 0.0042 \, C \][/tex]
[tex]\[ q_2 = -0.0050 \, C \][/tex]
[tex]\[ r = 0.0030 \, m \][/tex]

2. Substitute the values into the formula:
[tex]\[ F_e = \frac{(9.00 \times 10^9) \cdot (0.0042) \cdot (-0.0050)}{(0.0030)^2} \][/tex]

3. Calculate the numerator:
[tex]\[ (9.00 \times 10^9) \cdot (0.0042) \cdot (-0.0050) = 9.00 \times 10^9 \times -2.1 \times 10^{-5} = -1.89 \times 10^5 \, N \cdot m^2 / C^2 \][/tex]

4. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[ (0.0030 \, m)^2 = 0.000009 \, m^2 = 9 \times 10^{-6} \, m^2 \][/tex]

5. Divide the numerator by the denominator:
[tex]\[ F_e = \frac{-1.89 \times 10^5}{9 \times 10^{-6}} = -2.1 \times 10^{10} \, N \][/tex]

### Conclusion:
The electric force acting between the two charges is:

[tex]\[ -2.1 \times 10^{10} \, N \][/tex]

### Answer:
B. [tex]\(-2.1 \times 10^{10} \, N\)[/tex]