Two point charges, [math: q1=3.8×10−6C] and [math: q2=5.1×10−6C] , are placed 8.7 meters apart in a vacuum. Calculate the magnitude of the electrostatic force between them in Newtons.Give your answer in standard form to 2 decimal places using the following format without units:[math: 1.22×10−4]must be written as [ 1.22e-4 ]

Question

BlackTom AI · Accepted Answer

Coulomb’s law gives F = k q1 q2 / r^2, so use k = 8.98755179×10^9 N·m^2/C^2, q1 = 3.8×10^−6 C, q2 = 5.1×10^−6 C, and r = 8.7 m.

1) Relevant concepts:
- Coulomb’s law for electrostatic force between point charges.
- Use the vacuum permittivity constant incorporated in k = 8.98755179×10^9 N·m^2/C^2.
- Keep track of powers of ten and square the separation r.

2) Step-by-step reasoning and intermediate arithmetic:
- Compute the product of charges:
  q1·q2 = (3.8×10^−6)(5.1×10^−6) = 3.8×5.1 ×10^(−12) = 19.38×10^(−12) = 1.938×10^(−11) C^2.
- Multiply by k:
  k·(q1·q2) = (8.98755179×10^9)(1.938×10^(−11)) 
             = 8.98755179 × 1.938 ×10^(9−11)
             = 17.4181036 ×10^(−2)
             = 0.174181036 (in N·m^2).
- Square the distance:
  r^2 = (8.7 m)^2 = 75.69 m^2.
- Divide to get the force:
  F = 0.174181036 / 75.69 ≈ 0.0023010 N.

3) Final formatting and rounding:
- In standard form with two decimals in the mantissa: F ≈ 2.30×10^(−3) N, written as 2.30e-3 (no units as requested).

4) Why this is correct:
- All steps follow Coulomb’s law with correct constant and careful handling of powers of ten and r^2; the numeric division yields ~0.002301 N which rounds to 2.30e-3.

5) Common incorrect answers and why they are wrong:
- A value like 2.30e-6 would indicate a powers-of-ten error (forgetting the 10^3 factor from k or mis-multiplying exponents). 
- A larger value like 2.30e-2 would result from omitting r^2 in the denominator or using an incorrect (much smaller) r.
- Using k = 9×10^9 without proper significant-figure handling would give a similar result but might slightly change rounding; here the precise arithmetic yields 2.30e-3.

Conclusion: The calculated magnitude of the electrostatic force is 2.30e-3, which matches the provided answer.

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