Sure, let's solve the problem step-by-step:
### Step 1: Understand the Problem
We are asked to find the force between two charges. We have the following information:
- [tex]\( q1 = +3 \)[/tex] Coulombs
- [tex]\( q2 = -5 \)[/tex] Coulombs
- Distance ([tex]\( r \)[/tex]) between the charges is 1000 meters
### Step 2: Coulomb's Law
Coulomb's Law gives the force ([tex]\( F \)[/tex]) between two point charges as:
[tex]\[ F = k \cdot \frac{|q1 \cdot q2|}{r^2} \][/tex]
where:
- [tex]\( k \)[/tex] is the Coulomb constant [tex]\(( k \approx 8.9875517873681764 \times 10^9 \, \text{Nm}^2/\text{C}^2 ) \)[/tex]
- [tex]\( q1 \)[/tex] and [tex]\( q2 \)[/tex] are the magnitudes of the two charges
- [tex]\( r \)[/tex] is the distance between the charges
### Step 3: Substitute the Values
Substituting the given values into the formula:
1. Charges: [tex]\( +3 \)[/tex] Coulombs and [tex]\( -5 \)[/tex] Coulombs.
2. Distance: [tex]\( 1000 \)[/tex] meters.
Let's calculate the force:
[tex]\[ F = 8.9875517873681764 \times 10^9 \cdot \frac{|3 \cdot -5|}{1000^2} \][/tex]
### Step 4: Simplify the Calculation
- Calculate the product of the charges: [tex]\( 3 \times (-5) = -15 \)[/tex] (The magnitude is 15)
- The absolute value is: [tex]\( | -15 | = 15 \)[/tex]
- Square of the distance: [tex]\( 1000^2 = 1000000 \)[/tex]
Now substitute these back into the formula:
[tex]\[ F = 8.9875517873681764 \times 10^9 \cdot \frac{15}{1000000} \][/tex]
### Step 5: Calculate the Final Force
[tex]\[ F = 8.9875517873681764 \times 10^9 \cdot 1.5 \times 10^{-5} \][/tex]
[tex]\[ F = 13.481327681052264 \times 10^4 \][/tex]
[tex]\[ F \approx 134813.2768 \, \text{N} \][/tex]
### Step 6: Explanations and Answer Verification
Now, analyzing the numerical result approximately:
[tex]\[ 134813.2768 \, \text{N} \approx 135,000 \, \text{N} \][/tex]
We know the charges are opposite (one positive, one negative), meaning they attract each other.
### Conclusion:
So, the force between the two charges 1000 meters apart is approximately:
[tex]\[ F = 135,000 \, \text{N} \][/tex]
Hence, the correct option is:
d. 135,000 N towards each other
