Sure, let's solve this problem step-by-step.
Step 1: Identify given variables
- Source charge, \( q = 3 \times 10^{-6} \) C (Converting \( 3 \mu C \) to Coulombs)
- Electric field, \( E = 2.86 \times 10^5 \) N/C
- Coulomb's constant, \( k = 8.99 \times 10^9 \) N·m²/C²
Step 2: Understand the relationship
The electric field \( E \) created by a point charge \( q \) at a distance \( r \) is given by:
[tex]\[ E = \frac{k \cdot |q|}{r^2} \][/tex]
Step 3: Rearrange the equation to solve for \( r \)
[tex]\[ r^2 = \frac{k \cdot |q|}{E} \][/tex]
[tex]\[ r = \sqrt{\frac{k \cdot |q|}{E}} \][/tex]
Step 4: Substitute the known values into the equation
[tex]\[ r = \sqrt{\frac{(8.99 \times 10^9) \cdot (3 \times 10^{-6})}{2.86 \times 10^5}} \][/tex]
Step 5: Perform the calculations
First, calculate the numerator inside the square root:
[tex]\[ (8.99 \times 10^9) \cdot (3 \times 10^{-6}) = 26.97 \times 10^{3} \][/tex]
Then, divide by the electric field:
[tex]\[ \frac{26.97 \times 10^{3}}{2.86 \times 10^5} \approx 0.09435 \][/tex]
Finally, take the square root of this value:
[tex]\[ r = \sqrt{0.09435} \approx 0.30708418927176845 \][/tex]
Step 6: Round the result to the nearest hundredth
[tex]\[ r \approx 0.31 \][/tex]
So, the distance of the test charge from the source charge is approximately 0.31 meters to the nearest hundredth.
