Terry W. answered • 07/26/18
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In the context of this question, which I understand to mean that both conductors are carrying a constant (and presumably equal) current, I can only assume (see note below) that you are asking for the set of points where the net magnetic field generated by the current in the conductors is zero.
As a side note, magnetic induction refers to the production of an electromotive force (a voltage) in a conductor due to a changing magnetic field. Since your question makes no reference to a magnetic source other than the conductors themselves, I make the assumption that you are asking about magnetic field generated by a current instead. If that's not the case, I'm happy to amend my answer
With that out of the way, let's think about conceptually where magnetic fields from 2 perpendicular wires would be zero. The magnetic fields would have to exactly cancel each other out which means the magnitude of each would need to be equal while the direction would have to be exactly opposite. The latter constraint on the direction implies that the magnetic field vectors would need to be in the same plane whilst having opposite directions.
With those constraints in mind, we can try to visualize a portion of the magnetic fields by picturing 2 hollow pipes perpendicular to each other meeting at a junction (this would be a lot easier if pictures could be posted but I'll try to describe it). If you were to join the 2 pipes by welding them, what would the weld lines look like? It would look like 2 intersecting rings forming a X shape (looking from above) kind of like the frame of a ball.
Those 2 rings represent the set of points in 3D space where the magnitudes of the magnetic field from each wire is equal for a given distance (radius) from each wire. If you factored in every possible distance from 0 to infinity, you can visualize that as sets of concentric rings expanding outwards.
OK, so now that we've identified the places where magnitudes are equal, let's look at direction. Let's take just 1 set of 2 perpendicular rings and look at the directions of the magnetic field vectors from each wire on each ring. For sake of simplicity, for each ring, we only need to look at 4 points on the ring - 12 o'clock, 3 o'clock, 6 o'clock, and 9 o'clock. In which directions are the field lines pointed at each point? If you set a current direction in each wire (doesn't matter which way as long as you keep it consistent) and apply the right hand rule, you'll find that on both rings, the 12 o'clock and 6 o'clock positions are where the vectors are exactly at right angles and the 3 and 9 o'clock positions are where the vectors are either both in the same direction on 1 ring or in opposite directions on the other ring.
So on only 1 ring is there a set of points (2 to be exact) where the magnitudes of the field lines are equal AND the directions of the vectors are opposite. Now remember that each set of 2 rings represents a fixed distance from the wire where field line magnitudes are equal. So you need to account for all possible distances which are just concentric sets of 2 intersecting rings. So extrapolating from your 1 fixed distance case, you see that the set of all points where the field lines cancel is a diagonal line.
The answer therefore is that the set of points where the magnetic fields from 2 perpendicular wires carrying identical currents sum to zero is a 45 degree diagonal line in the plane of the wires that runs through their point of intersection. Which quadrants the line runs through depends on the directions of the current in the 2 wires.
2 other side notes:
- The angle of the diagonal line depends on the magnitudes of the currents in the wires. If the currents are equal, then the angle is 45 degrees. If they are not equal, then the angle will change
- Based on symmetry, the other symmetrical diagonal line running through the other 2 quadrants will be the set of points where the magnetic field is the maximum (exactly additive).
