Two long, parallel wires carry current in the x-y plane. One wire carries 30 A to the left along the x-axis. The other carries 50 A to the right along a parallel line at y = 0.28 m. At what y-axis position in meters is the magnetic field equal to zero?

μ₀I₁/2πy= μ₀I₂/2π(0.28-y)

I₁(0.28-y)= I₂y
30(0.28-y)=50y
80y=8.4
y=8.4/80=0.105 m

To find the y-axis position where the magnetic field is equal to zero, we can use the Biot-Savart law, which states that the magnetic field at a point due to a current-carrying wire can be calculated by integrating the contributions from each infinitesimally small segment of the wire.

Let's break down the problem step by step:

1. Determine the direction of the magnetic field created by each wire segment:
- The wire carrying current to the left (along the x-axis) induces a magnetic field that loops around the wire in counterclockwise fashion (using the right-hand rule).
- The wire carrying current to the right (along the parallel line at y = 0.28 m) induces a magnetic field that loops around the wire in clockwise fashion (using the right-hand rule).

2. Calculate the magnetic field contribution from each wire segment at a point (x, y):
- For the wire carrying current to the left:
- The magnetic field contribution at point P(x, y) due to a small segment of the wire, dl, is given by:
dB_left = (μ₀ / 4π) * (30 A) / r_left² * dl × sin(θ_left)
where μ₀ is the permeability of free space, r_left is the distance from the wire segment to point P, dl is the length of the wire segment, and θ_left is the angle between the wire segment and the line connecting the wire to point P.

- For the wire carrying current to the right:
- The magnetic field contribution at point P(x, y) due to a small segment of the wire, dl, is given by:
dB_right = (μ₀ / 4π) * (-50 A) / r_right² * dl × sin(θ_right)
where μ₀ is the permeability of free space, r_right is the distance from the wire segment to point P, dl is the length of the wire segment, and θ_right is the angle between the wire segment and the line connecting the wire to point P.

3. Integrate the magnetic field contributions from each wire segment along the respective wires to find the magnetic field at point P(x, y). Since we are interested in finding the y-axis position where the magnetic field is zero, we can set up the equation by setting the total magnetic field equal to zero:
B = Σ(dB_left) + Σ(dB_right) = 0
Note: The sum is taken over all wire segments.

4. Solve the equation to find the value of y where the magnetic field is zero. This can be done numerically or using software tools like Mathematica, Matlab, or Python.

After solving the equation, you will obtain the y-value where the magnetic field is equal to zero.