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What is the electrical force between [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex]?

Recall that [tex]\( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)[/tex].

Given:
- [tex]\( q_1 = +6 \, \text{C} \)[/tex]
- [tex]\( q_2 = -4 \, \text{C} \)[/tex]
- [tex]\( r = 3 \, \text{m} \)[/tex]

Possible answers:
A. [tex]\( 4.3 \times 10^{10} \, \text{N} \)[/tex]
B. [tex]\( 3.5 \times 10^{10} \, \text{N} \)[/tex]
C. [tex]\( -5.4 \times 10^{10} \, \text{N} \)[/tex]
D. [tex]\( -5.8 \times 10^{10} \, \text{N} \)[/tex]

Sagot :

To determine the electrical force between charges [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex], we utilize Coulomb's Law. Coulomb's Law states that the magnitude of the electrical force [tex]\( F \)[/tex] between two charges is given by:

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

Here:
- [tex]\( k \)[/tex] is the Coulomb constant, [tex]\( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)[/tex]
- [tex]\( q_1 \)[/tex] is the magnitude of the first charge, [tex]\( q_1 = 6 \, \text{C} \)[/tex]
- [tex]\( q_2 \)[/tex] is the magnitude of the second charge, [tex]\( q_2 = -4 \, \text{C} \)[/tex]
- [tex]\( r \)[/tex] is the distance between the charges, [tex]\( r = 3 \, \text{m} \)[/tex]

First, we note that Coulomb's Law uses the absolute value of the product of charges, thus:

[tex]\[ |q_1 q_2| = |6 \cdot (-4)| = | -24 | = 24 \, \text{C}^2 \][/tex]

Next, we substitute these values into the equation:

[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \cdot \frac{24 \, \text{C}^2}{(3 \, \text{m})^2} \][/tex]

We calculate the denominator first:

[tex]\[ r^2 = (3 \, \text{m})^2 = 9 \, \text{m}^2 \][/tex]

Now, substituting this back into the equation:

[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \cdot \frac{24 \, \text{C}^2}{9 \, \text{m}^2} \][/tex]

Simplify the fraction:

[tex]\[ \frac{24 \, \text{C}^2}{9 \, \text{m}^2} = \frac{24}{9} = 2.6667 \, \text{C}^2/\text{m}^2 \][/tex]

Then:

[tex]\[ F = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \cdot 2.6667 \, \text{C}^2/\text{m}^2 \][/tex]

Multiplying these together gives us:

[tex]\[ F \approx 2.3973333333 \times 10^{10} \, \text{N} \][/tex]

Therefore, the electrical force between [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] is approximately [tex]\( 2.397 \times 10^{10} \, \text{N} \)[/tex], which can be written as:

[tex]\[ F \approx 23973333333.333332 \, \text{N} \][/tex]

Among the given options, this result does not exactly match any provided choices, so there might be a mistake or a different context in the options list. However, the calculated value stands as accurate for the data given.