Two long parallel wires carry currents of 20 A and 5.0 A in opposite directions. The wires are separated by 0.20 m. At what point are the contributions to the magnetic field from the two wires the same?

Question

BlackTom AI · Accepted Answer

For a long straight wire the magnetic field magnitude is B = μ0 I / (2π r). The μ0/2π factor cancels when equating two fields, so the condition for equal magnitudes is I1 / r1 = I2 / r2.

Label the distance from the 20 A wire to the point of interest x (r1 = x). The wires are separated by d = 0.20 m, so the distance from the 5.0 A wire to the same point is r2 = d − x = 0.20 − x. Set the magnitudes equal:

I1 / r1 = I2 / r2
20 A / x = 5.0 A / (0.20 − x)

Solve step by step:
20(0.20 − x) = 5.0 x
4.0 − 20 x = 5 x       (20 × 0.20 = 4.0)
4.0 = 25 x
x = 4.0 / 25 = 0.16 m

Thus the point where the two wires produce equal magnetic field magnitudes lies 0.16 m from the 20 A wire (which places it 0.04 m from the 5.0 A wire). This location is closer to the weaker (5 A) wire, as expected: a stronger current must be farther away for its field to match the weaker current’s field.

Why other choices are incorrect:
- 0.040 m from the 20 A wire is actually the distance from the 5 A wire (d − 0.16), not from the 20 A wire, and would not satisfy I1 / r1 = I2 / r2 when r1 = 0.04 m.
- 0.080 m and 0.12 m do not satisfy the equality 20/(x) = 5/(0.20 − x); substituting either value produces unequal left/right sides.

Confirming: the correct answer is 0.16 m from the 20 A wire.

Two long parallel wires carry currents of 20 A and 5.0 A in opposite directions. The wires are separated by 0.20 m. At what point are the contributions to the magnetic field from the two wires the same?单项选择题

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