To determine the distance between the source charge and the test charge, we can use the formula for the electric field due to a point charge:
[tex]\[ E = \frac{k \cdot |q|}{r^2} \][/tex]
Where:
- [tex]\( E \)[/tex] is the electric field,
- [tex]\( k \)[/tex] is Coulomb's constant,
- [tex]\( q \)[/tex] is the source charge,
- [tex]\( r \)[/tex] is the distance from the source charge.
Given values:
- Source charge, [tex]\( q = 3 \times 10^{-6} \ \text{C} \)[/tex] (i.e., 3 microCoulombs converted to Coulombs),
- Electric field, [tex]\( E = 2.86 \times 10^5 \ \text{N/C} \)[/tex],
- Coulomb's constant, [tex]\( k = 8.99 \times 10^9 \ \text{N} \cdot \text{m}^2/\text{C}^2 \)[/tex].
Our goal is to solve for [tex]\( r \)[/tex]. First, re-arrange the electric field equation to solve for [tex]\( r \)[/tex]:
[tex]\[ r^2 = \frac{k \cdot |q|}{E} \][/tex]
Next, take the square root of both sides to isolate [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{k \cdot |q|}{E}} \][/tex]
Substitute in the given values:
[tex]\[ r = \sqrt{\frac{8.99 \times 10^9 \ \text{N} \cdot \text{m}^2/\text{C}^2 \cdot 3 \times 10^{-6} \ \text{C}}{2.86 \times 10^5 \ \text{N/C}}} \][/tex]
Carefully calculate the value inside the square root:
[tex]\[ r = \sqrt{\frac{8.99 \times 10^9 \cdot 3 \times 10^{-6}}{2.86 \times 10^5}} \][/tex]
[tex]\[ r = \sqrt{\frac{26.97 \times 10^3}{2.86 \times 10^5}} \][/tex]
[tex]\[ r = \sqrt{0.094358 \text{ m}^2} \][/tex]
[tex]\[ r = 0.307084 \ \text{m} \][/tex]
To the nearest hundredth, the distance [tex]\( r \)[/tex] is:
[tex]\[ r = 0.31 \ \text{m} \][/tex]
Thus, the distance of the test charge from the source charge is [tex]\( \boxed{0.31} \ \text{meters} \)[/tex].
