Certainly! Let's analyze each of the given formulas to determine which one correctly represents the strength of an electric field, [tex]\( E \)[/tex], at a distance from a known source charge.
The concept of the electric field, [tex]\( E \)[/tex], is defined as the force per unit charge. Mathematically, this relationship is expressed as:
[tex]\[ E = \frac{F}{q} \][/tex]
where:
- [tex]\( E \)[/tex] is the electric field strength,
- [tex]\( F \)[/tex] is the force experienced by the test charge,
- [tex]\( q \)[/tex] is the test charge itself.
From this definition, we can now evaluate each of the given formulas:
1. [tex]\( E = \frac{F}{Q d} \)[/tex]:
This formula suggests that [tex]\( E \)[/tex] is the force divided by the product of charge [tex]\( Q \)[/tex] and distance [tex]\( d \)[/tex]. This does not match the definition of electric field strength as it incorrectly introduces an additional distance factor in the denominator.
2. [tex]\( E = \frac{k a}{d} \)[/tex]:
Here, [tex]\( E \)[/tex] is represented as a product of constant [tex]\( k \)[/tex] and acceleration [tex]\( a \)[/tex] divided by distance [tex]\( d \)[/tex]. This form does not align with the concept of an electric field derived from force per unit charge.
3. [tex]\( E = \frac{k q}{\sigma^2} \)[/tex]:
In this formula, [tex]\( E \)[/tex] is given as a product of constant [tex]\( k \)[/tex] and charge [tex]\( q \)[/tex] divided by [tex]\( \sigma^2 \)[/tex]. This interpretation does not fit the basic principles or units involved in defining an electric field.
4. [tex]\( E = \frac{F_o}{d} \)[/tex]:
This formula suggests an electric field strength as force [tex]\( F_o \)[/tex] divided by distance [tex]\( d \)[/tex]. While this might relate to a specific case, it does not universally represent the fundamental form of an electric field strength.
After carefully considering the definitions and meanings of the symbols involved, we arrive at the conclusion:
[tex]\[ E = \frac{F}{Q} \][/tex]
Given these considerations, the correct formula for the electric field strength, [tex]\( E \)[/tex], at a distance from a known source charge should be aligned with the fundamental definition:
However, among the given choices, none of them perfectly match the foundational formula of [tex]\( E = \frac{F}{q} \)[/tex]. The closest and most exact representation based on correct usage is indeed:
[tex]\[ E = \frac{F}{Q d} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{1} \][/tex]
