To find the distance \( r \) from the source charge to the test charge, we can use the formula for the electric field \( E \) generated by a point charge \( Q \):
\[ E = \frac{k |Q|}{r^2} \]
where:
- \( E \) is the electric field (in N/C),
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 , \text{N m}^2/\text{C}^2 \)),
- \( |Q| \) is the magnitude of the source charge (in C),
- \( r \) is the distance between the source charge and the test charge (in m).
We know:
- The source charge \( Q = 3 , \mu\text{C} = 3 \times 10^{-6} , \text{C} \)
- The electric field \( E = 2.86 \times 10^5 , \text{N/C} \)
We can rearrange the formula to solve for the distance \( r \):
\[ r^2 = \frac{k |Q|}{E} \]
To isolate \( r \), we take the square root:
\[ r = \sqrt{\frac{k |Q|}{E}} \]
Now, plug in the values:
\[ r = \sqrt{\frac{(8.99 \times 10^9 , \text{N m}^2/\text{C}^2)(3 \times 10^{-6} , \text{C})}{2.86 \times 10^5 , \text{N/C}}} \]
First, we calculate the numerator:
\[ (8.99 \times 10^9)(3 \times 10^{-6}) = 26.97 \times 10^3 = 2.697 \times 10^4 , \text{N m}^2/\text{C} \]
Now calculate \( r^2 \):
\[ r^2 = \frac{2.697 \times 10^4}{2.86 \times 10^5} \]
Calculating the division:
\[ r^2 = \frac{2.697 \times 10^4}{2.86 \times 10^5} = \frac{2.697}{2.86} \times 10^{-1} \approx 0.09432 , \text{m}^2 \]
Now take the square root to find \( r \):
\[ r \approx \sqrt{0.09432} \approx 0.307 , \text{m} \]
Rounding to the nearest hundredth:
\[ r \approx 0.31 , \text{m} \]
Thus, the distance of the test charge from the source charge is approximately \(\boxed{0.31}\) meters.