To find the correct equation for calculating the electrical force, [tex]\( F_e \)[/tex], between two charges, we use Coulomb's law. The law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.
Given the following options:
1. [tex]\( F_e = k \frac{q_1 q_2}{d} \)[/tex]
2. [tex]\( F_e = \frac{q_1 q_2}{d^2} \)[/tex]
3. [tex]\( F_e = k \frac{q_1 q_2}{d^2} \)[/tex]
4. [tex]\( F_e = k\left(\frac{q_1 q_2}{d}\right)^2 \)[/tex]
We analyze each option in detail:
1. [tex]\( F_e = k \frac{q_1 q_2}{d} \)[/tex]
- This equation suggests that the force is inversely proportional to the distance (not its square). However, Coulomb's law specifies that the force is inversely proportional to the square of the distance, so this option is incorrect.
2. [tex]\( F_e = \frac{q_1 q_2}{d^2} \)[/tex]
- This equation lacks the proportionality constant [tex]\( k \)[/tex], which is critical in Coulomb's law. Hence, this option is incomplete and incorrect.
3. [tex]\( F_e = k \frac{q_1 q_2}{d^2} \)[/tex]
- This equation correctly includes both the proportionality constant [tex]\( k \)[/tex], the product of the charges, and the inverse square relationship with the distance. This equation matches Coulomb's law precisely.
4. [tex]\( F_e = k\left(\frac{q_1 q_2}{d}\right)^2 \)[/tex]
- This equation suggests that the force is proportional to the square of the quotient of the product of charges and the distance, which is incorrect according to Coulomb's law.
Therefore, the correct equation for calculating the electrical force [tex]\( F_e \)[/tex] between two charges is:
[tex]\[ F_e = k \frac{q_1 q_2}{d^2} \][/tex]
So the correct option is:
[tex]\[ \boxed{3} \][/tex]
