Sure, let’s go through the detailed solution for determining the correct formula for the strength of an electric field, [tex]\( E \)[/tex], at a distance [tex]\( d \)[/tex] from a known source charge [tex]\( q \)[/tex].
According to Coulomb's law, the electric field [tex]\( E \)[/tex] produced by a point charge [tex]\( q \)[/tex] at a distance [tex]\( d \)[/tex] away from the charge is given by:
[tex]\[
E = \frac{k q}{d^2}
\][/tex]
where:
- [tex]\( E \)[/tex] is the electric field,
- [tex]\( k \)[/tex] is Coulomb's constant (approximately [tex]\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex]),
- [tex]\( q \)[/tex] is the source charge,
- [tex]\( d \)[/tex] is the distance from the charge.
Let's analyze the given options:
1. [tex]\( E = \frac{F_2}{q d} \)[/tex]
This formula does not match Coulomb's law. The dimensions do not appropriately simplify to that of an electric field.
2. [tex]\( E = \frac{k a}{d} \)[/tex]
This formula is incorrect as it introduces an acceleration [tex]\( a \)[/tex] which does not relate directly to the electric field equation derived from Coulomb's law.
3. [tex]\( E = \frac{k q}{d^2} \)[/tex]
This formula matches Coulomb's law precisely. It correctly represents the relationship between the electric field, the source charge, and the distance from the charge according to Coulomb's law.
4. [tex]\( E = \frac{F_e}{d} \)[/tex]
This formula does not properly account for the dependency of the electric field on the inverse square of the distance from the charge, which is essential in Coulomb's law.
Therefore, the correct answer is:
[tex]\[
E = \frac{k q}{d^2}
\][/tex]
So, option 3 is the correct formula for the strength of an electric field, [tex]\( E \)[/tex], at a distance [tex]\( d \)[/tex] from a known source charge [tex]\( q \)[/tex].
