To determine the correct formula for the strength of an electric field [tex]\( E \)[/tex] at a distance from a known source charge, let's examine each of the provided options and assess them based on principles of electrostatics.
1. Option 1: [tex]\( E = \frac{F_e}{q d} \)[/tex]
This formula is incorrect because it misuses Coulomb's Law. Coulomb's Law gives [tex]\( F_e \)[/tex] for two charges, and the electric field is the force per unit charge, not divided by distance.
2. Option 2: [tex]\( E = \frac{k q}{d} \)[/tex]
This formula is partially correct but lacks completeness. It should include the square of the distance, as per Coulomb’s Law, where the electric field due to a point charge decreases with the square of the distance.
3. Option 3: [tex]\( E = \frac{k q}{d^2} \)[/tex]
This is the correct formula. According to Coulomb’s Law, the electric field [tex]\( E \)[/tex] created by a point charge [tex]\( q \)[/tex] at a distance [tex]\( d \)[/tex] in a vacuum is given by:
[tex]\[
E = \frac{k q}{d^2}
\][/tex]
where [tex]\( k \)[/tex] is Coulomb's constant ([tex]\( k \approx 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex]). This formula accurately represents how the strength of the electric field diminishes with the square of the distance from the source charge.
4. Option 4: [tex]\( E = \frac{F_e}{d} \)[/tex]
This formula is invalid because it misrepresents the relationship between force and electric field. The electric field is defined as the force per unit charge, which does not involve distance in this manner.
Given the analysis of all the options, the correct formula that represents the strength of an electric field [tex]\( E \)[/tex] at a distance [tex]\( d \)[/tex] from a known source charge [tex]\( q \)[/tex] is:
[tex]\[
E = \frac{k q}{d^2}
\][/tex]
Hence, the answer is option 3.
