A source charge of [tex]$3 \mu C[tex]$[/tex] generates an electric field of [tex]$[/tex]2.86 \times 10^5 \, N/C$[/tex] at the location of a test charge. What is the distance, to the nearest hundredth, of the test charge from the source charge?

[tex]k = 8.99 \times 10^9 \, N \cdot m^2/C^2[/tex]

[tex]\boxed{\text{m}}[/tex]



Answer :

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.