Answer:
Approximately
, 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
, which means that the first wire (with current
) is underneath the second wire (with current
.) Let
denote the
-coordinate (in
) of a point of interest. The two wires partition this region into three parts:
- The region under the first wire, where
; - The region between the first and the second wire, where
; and - The region above the second wire, where
.
Apply the right-hand rule to find the direction of the magnetic field in each of the three regions. Assume that
points to the left while
points to the right. Let
and
denote the magnetic field due to
and
, respectively.
- In the region below the first wire,
points out of the
-plane while
points into the
-plane. - In the region between the two wires, both
and
point into the
-plane. - In the region above the second wire,
points into the
-plane while
points out of the
-plane.
The (net) magnetic field on this plane would be zero only in regions where
and
points in opposite directions. That rules out the region between the two wires.
At a distance of
away from a wire with current
and infinite length, the formula for the magnitude of the magnetic field
due to that wire is:
,
where
is the permeability of the space between the wire and the point of interest (should be constant.) The exact value of
does not affect the answer to this question, as long as it is constant throughout this region.
Note that the value of
in this formula is supposed to be positive. Let
and
denote the distance between the point of interest and the two wires, respectively.
- In the region under the first wire,
, while
. - In the region above the second wire,
, while
.
Make sure that given the corresponding range of
, these distances are all positive.
- Strength of the magnetic field due to the first wire:
. - Strength of the magnetic field due to the second wire:
.
For the net magnetic field to be zero at a certain
-value, the strength of the two magnetic fields at that point should match. That is:
.
.
Simplify this equation:
.
- In the region under the first wire (where
,) this equation becomes
. - In the region above the second wire (where
,) this equation becomes
.
These two equations give the same result:
. However, based on the respective assumptions on the value of
, this value corresponds to the region above the second wire.