A charge of 2 X 10-7 C is acted upon by a force of 0.1N. Determine the distance to the other charge of 4.5 X 10-7 C with both the charges in vacuum.

Question

BlackTom AI · Accepted Answer

Coulomb's law gives F = k q1 q2 / r^2, so r = sqrt( k q1 q2 / F ).

1) Relevant concepts:
- Coulomb's law for point charges in vacuum: F = k q1 q2 / r^2, where k = 8.99 × 10^9 N·m^2/C^2 (k = 1/(4πϵ0)).
- Solve for distance r by algebraic rearrangement and taking a square root.

2) Step-by-step reasoning and algebra:
Start with F = k q1 q2 / r^2.
Rearrange: r^2 = k q1 q2 / F, therefore r = sqrt( k q1 q2 / F ).

3) Numerical calculation with intermediate steps:
- Given q1 = 2.0 × 10^-7 C, q2 = 4.5 × 10^-7 C, F = 0.1 N, k = 8.99 × 10^9 N·m^2/C^2.
- Compute product q1·q2 = (2.0 × 10^-7)·(4.5 × 10^-7) = 9.0 × 10^-14 C^2.
- Multiply by k: k·(q1 q2) = 8.99 × 10^9 · 9.0 × 10^-14 = (8.99·9.0) × 10^(9-14) = 80.91 × 10^-5 = 8.091 × 10^-4 (units: N·m^2).
- Divide by F: (k q1 q2)/F = 8.091 × 10^-4 / 0.1 = 8.091 × 10^-3 m^2.
- Take square root: r = sqrt(8.091 × 10^-3) ≈ sqrt(8.091) × 10^-1.5 ≈ 2.844 × 0.0316228 ≈ 0.08999 m ≈ 0.09 m.

4) Why this is correct:
The calculation directly applies Coulomb's law with standard k and the given charges and force. The algebra and arithmetic yield r ≈ 0.09 m, which matches the physical expectation that modest microcoulomb charges at ~centimeter scale produce a force of 0.1 N.

5) Why the other options are incorrect:
- 0.03 m, 0.05 m, 0.07 m are all significantly smaller than 0.09 m; smaller separations would produce a larger force than 0.1 N for these charges, so they contradict the given force magnitude. The computed value (0.09 m) is the only one consistent with Coulomb's law and the provided numbers.

6) Conclusion:
The correct distance is 0.09 m, matching answer choice b.

A charge of 2 X 10-7 C is acted upon by a force of 0.1N. Determine the distance to the other charge of 4.5 X 10-7 C with both the charges in vacuum.单项选择题

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