Two point charges are placed along a horizontal axis with the following values and positions: +3.0 µC at x = 0 cm and -6.0 µC at x = 20 cm. At what point along the x axis is the electric field zero?

Question

Two point charges are placed along a horizontal axis with the following values and positions: +3.0 µC at x = 0 cm and -6.0 µC at x = 20 cm.  At what point along the x axis is the electric field zero?

BlackTom AI · Accepted Answer

Because the electric field of a point charge has magnitude E = k|q|/r^2 and points away from positive charges and toward negative charges, set the magnitudes of the fields from the two charges equal where their directions are opposite. Positions and signs: q1 = +3.0 μC at x = 0 cm, q2 = −6.0 μC at x = 20 cm. Consider possible regions: - Between the charges (0 < x < 20): the +3 μC produces a field to the right, the −6 μC also produces a field to the right (field points toward the negative), so fields add and cannot cancel → no zero here. - To the right of both (x > 20): fields are opposite, but solving gives a root inside (0,20) so not physical here. - To the left of both (x < 0): fields are opposite and a valid solution exists. Let x = −d (d > 0) be the point left of the origin. Distances to charges: r1 = d (to +3 μC), r2 = 20 + d (to −6 μC). Equal magnitudes condition: k*(3.0)/d^2 = k*(6.0)/(20 + d)^2 Cancel k and simplify: 3/d^2 = 6/(20 + d)^2 (20 + d)^2 = 2 d^2 Expand: 400 + 40d + d^2 = 2d^2 Rearrange: 400 + 40d − d^2 = 0 → d^2 − 40d − 400 = 0 Solve quadratic: d = [40 ± sqrt(40^2 + 4·400)]/2 = [40 ± sqrt(3200)]/2 sqrt(3200) ≈ 56.5685, so d = (40 + 56.5685)/2 ≈ 48.284 cm (positive root; the other root is negative and unphysical). Thus x = −d ≈ −48.28 cm, which rounds to −48 cm. Why other choices fail: 0 cm is the location of a charge (field undefined/infinite); 8 cm (~8.28 cm) is the quadratic’s other root but lies between charges where fields point the same way so cancellation cannot occur; 69 cm and −12 cm do not satisfy the equality of field magnitudes. Therefore the electric field is zero at approximately −48 cm, matching the provided answer: −48cm.

Two point charges are placed along a horizontal axis with the following values and positions: +3.0 µC at x = 0 cm and -6.0 µC at x = 20 cm. At what point along the x axis is the electric field zero?单项选择题

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