Question

Two wires carry current I1 = 47 A and I2 = 29 A in the opposite directions parallel to the x-axis at y1 = 9 cm and y2 = 15 cm. Where on the y-axis (in cm) is the magnetic field zero?

Answer 1

Approximately
`25\; \rm cm`, assuming that the permeability between the two wires is constant, and that the two wires are of infinite lengths.

Step-by-step explanation:

Note that
`y_1 < y_2`, which means that the first wire (with current
`I_1`) is underneath the second wire (with current
`I_2`.) Let
`y` denote the
`y`-coordinate (in
`\rm cm`) of a point of interest. The two wires partition this region into three parts:

• The region under the first wire, where
  `y < y_1`;
• The region between the first and the second wire, where
  `y_1 < y < y_2`; and
• The region above the second wire, where
  `y > y_2`.

Apply the right-hand rule to find the direction of the magnetic field in each of the three regions. Assume that
`I_1` points to the left while
`I_2` points to the right. Let
`B_1` and
`B_2` denote the magnetic field due to
`I_1` and
`I_2`, respectively.

• In the region below the first wire,
  `B_1` points out of the
  `x y`-plane while
  `B_2` points into the
  `x y`-plane.
• In the region between the two wires, both
  `B_1` and
  `B_2` point into the
  `x y`-plane.
• In the region above the second wire,
  `B_1` points into the
  `x y`-plane while
  `B_2` points out of the
  `x y`-plane.

The (net) magnetic field on this plane would be zero only in regions where
`B_1` and
`B_2` points in opposite directions. That rules out the region between the two wires.

At a distance of
`R` away from a wire with current
`I` and infinite length, the formula for the magnitude of the magnetic field
`B` due to that wire is:

`\displaystyle B = (\mu\, I)/(2\pi\, R)`,

where
`\mu` is the permeability of the space between the wire and the point of interest (should be constant.) The exact value of
`\mu` does not affect the answer to this question, as long as it is constant throughout this region.

Note that the value of
`R` in this formula is supposed to be positive. Let
`R_1` and
`R_2` denote the distance between the point of interest and the two wires, respectively.

• In the region under the first wire,
  `R_1 = y_1 - y = 9 - y`, while
  `R_2 = y_2 - y = 15 - y`.
• In the region above the second wire,
  `R_1 = y - 9`, while
  `R_2 = y - 15`.

Make sure that given the corresponding range of
`y`, these distances are all positive.

• Strength of the magnetic field due to the first wire:
  `\displaystyle B_1 = (\mu\, I_1)/(4\pi\, R_1)`.
• Strength of the magnetic field due to the second wire:
  `\displaystyle B_1 = (\mu\, I_2)/(4\pi\, R_2)`.

For the net magnetic field to be zero at a certain
`y`-value, the strength of the two magnetic fields at that point should match. That is:

`B_1 = B_2`.

`\displaystyle (\mu\, I_1)/(4\pi\, R_1) = (\mu\, I_2)/(4\pi\, R_2)`.

Simplify this equation:

`\displaystyle (I_1)/(R_1) = (I_2)/(R_2)`.

• In the region under the first wire (where
  `y < 9`,) this equation becomes
  `\displaystyle (47)/(9 - y) = (29)/(15 - y)`.
• In the region above the second wire (where
  `y > 15`,) this equation becomes
  `\displaystyle (47)/(y - 9) = (29)/(y - 15)`.

These two equations give the same result:
`y = 25`. However, based on the respective assumptions on the value of
`y`, this value corresponds to the region above the second wire.