Certainly! Here's the detailed, step-by-step solution to find the force between two equal charges of 1 Coulomb each, separated by a distance of 1 kilometer in air:
Step 1: Understand Coulomb's Law
Coulomb's Law describes the electrostatic force between two charged objects. The formula is given by:
[tex]\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the magnitude of the force between charges,
- [tex]\( k \)[/tex] is Coulomb's constant ([tex]\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)[/tex]),
- [tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex] are the magnitudes of the charges,
- [tex]\( r \)[/tex] is the distance between the charges.
Step 2: Identify the given values
In this problem:
- [tex]\( q_1 = 1 \, \text{C} \)[/tex]
- [tex]\( q_2 = 1 \, \text{C} \)[/tex]
- [tex]\( r = 1000 \, \text{m} \)[/tex] (since 1 kilometer is equivalent to 1000 meters)
- [tex]\( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)[/tex]
Step 3: Substitute the values into Coulomb's Law
Using the values provided:
[tex]\[ F = (8.99 \times 10^9) \frac{|1 \cdot 1|}{(1000)^2} \][/tex]
Step 4: Simplify the expression
First, simplify the denominator:
[tex]\[ (1000)^2 = 1000000 \][/tex]
Then simplify the fraction:
[tex]\[ \frac{|1 \cdot 1|}{1000000} = \frac{1}{1000000} \][/tex]
Now substitute back into the equation:
[tex]\[ F = (8.99 \times 10^9) \times \frac{1}{1000000} \][/tex]
Simplify the multiplication:
[tex]\[ F = \frac{8.99 \times 10^9}{10^6} = 8.99 \times 10^3 \][/tex]
Step 5: Convert the final value
[tex]\[ 8.99 \times 10^3 = 8990 \][/tex]
Therefore, the magnitude of the force between the two charges is:
[tex]\[ \boxed{8990 \, \text{N}} \][/tex]
This completes the solution. The force between two equal charges of 1 Coulomb each, separated by a distance of 1 kilometer in air, is 8990 Newtons.
