Two point charges of +3.0 μC and −7.0 μC are placed at x = 0 and x = 0.20 m. What is the magnitude of the electric field at the point midway between them?

Question

BlackTom AI · Accepted Answer

The electric field from a point charge is E = k|q|/r^2, directed away from a positive charge and toward a negative charge. Use k = 8.99 × 10^9 N·m^2/C^2.

1) Relevant concepts
- Superposition: the net field is the vector sum of fields from each charge.
- Magnitude from each charge uses E = k|q|/r^2.
- Directions: a positive charge produces a field pointing away; a negative charge produces a field pointing toward it.

2) Step-by-step reasoning and calculation
- Positions: q1 = +3.0 μC at x = 0, q2 = −7.0 μC at x = 0.20 m. Midpoint is at x = 0.10 m.
- Distance from midpoint to each charge: r = 0.10 m.

Compute field magnitudes from each charge:
- E1 from q1: E1 = k|q1|/r^2
  = (8.99 × 10^9 N·m^2/C^2)(3.0 × 10^−6 C) / (0.10 m)^2
  = (8.99 × 10^9)(3.0 × 10^−6)/0.01
  = (26.97 × 10^3)/0.01 = 2.697 × 10^6 N/C ≈ 2.70 × 10^6 N/C.

- E2 from q2: E2 = k|q2|/r^2
  = (8.99 × 10^9)(7.0 × 10^−6)/0.01
  = (62.93 × 10^3)/0.01 = 6.293 × 10^6 N/C ≈ 6.29 × 10^6 N/C.

3) Direction and superposition
- At the midpoint, E1 points to the right (away from +3.0 μC). E2 also points to the right (toward −7.0 μC). Since both are in the same direction, the magnitudes add.

4) Net magnitude
- E_total = E1 + E2 = 2.697 × 10^6 + 6.293 × 10^6 = 8.990 × 10^6 N/C ≈ 9.0 × 10^6 N/C.

5) Why the other options are incorrect
- 1.8 × 10^6, 3.6 × 10^6, and 4.5 × 10^6 N/C are much smaller; they would result from incorrect arithmetic (e.g., forgetting to divide by r^2 properly, using wrong distances, or not adding the contributions). Only 9.0 × 10^6 N/C matches the correct superposed result.

Conclusion: The magnitude of the electric field at the midpoint is 9.0 × 10^6 N/C, which matches the provided correct answer.

Two point charges of +3.0 μC and −7.0 μC are placed at x = 0 and x = 0.20 m. What is the magnitude of the electric field at the point midway between them?单项选择题

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