To determine the distance between the source charge and the test charge, we can use Coulomb's Law as it pertains to electric fields. The relationship is given by:
[tex]\[ E = k \frac{Q}{r^2} \][/tex]
where:
- [tex]\(E\)[/tex] is the electric field,
- [tex]\(k\)[/tex] is Coulomb's constant ( [tex]\(8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2} \)[/tex] ),
- [tex]\(Q\)[/tex] is the source charge ( [tex]\(3 \, \mu C = 3 \times 10^{-6} \, C\)[/tex] ), and
- [tex]\(r\)[/tex] is the distance we need to find.
We need to solve for [tex]\(r\)[/tex], the distance. Rearranging the equation to solve for [tex]\(r^2\)[/tex], we get:
[tex]\[ r^2 = k \frac{Q}{E} \][/tex]
Taking the square root of both sides, we get:
[tex]\[ r = \sqrt{ \frac{kQ}{E} } \][/tex]
Substituting in the given values:
[tex]\[ r = \sqrt{ \frac{8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2} \times 3 \times 10^{-6} \, C}{2.86 \times 10^5 \, \frac{N}{C}} } \][/tex]
Calculate the value inside the square root:
[tex]\[ r = \sqrt{ \frac{8.99 \times 10^9 \times 3 \times 10^{-6}}{2.86 \times 10^5} } \][/tex]
Divide the products in the numerator and the denominator:
[tex]\[ r = \sqrt{ \frac{26.97 \times 10^3}{2.86 \times 10^5} } \][/tex]
Simplify the exponent part:
[tex]\[ r = \sqrt{ \frac{26.97}{2.86 \times 10^2} } \][/tex]
[tex]\[ r = \sqrt{ \frac{26.97}{286} } \][/tex]
Calculate the fraction:
[tex]\[ r = \sqrt{ 0.09432 } \][/tex]
Finally, take the square root:
[tex]\[ r \approx 0.30708418927176845 \][/tex]
Rounding this to the nearest hundredth:
[tex]\[ r \approx 0.31 \, m \][/tex]
Thus, the distance of the test charge from the source charge, rounded to the nearest hundredth, is:
[tex]\[ \boxed{0.31} \, m \][/tex]
