Two long, parallel wires carry currents of one amp and three amps, respectively, in the direction shown in the figure. At which point — 𝐴, 𝐵, 𝐶, 𝐷, or 𝐸 — is the magnetic field zero?
Looking at this diagram, we see the two long straight wires, which carry currents of one amp and three amps. Each of these wires we know will create a magnetic field around it. And we want to solve for the location 𝐴, 𝐵, 𝐶, 𝐷, or 𝐸 at which the magnetic field in total is zero.
The magnetic field 𝐵 created by a long, straight current-carrying wire is equal to the permeability of free space times the current in the wire all divided by two 𝜋 times the distance from the wire, at which the field is being calculated. In this instance, we’re interested not just in the magnitude of the field, but also in its direction.
Considering the one amp current moving along the line marked 𝑥, if we use our right-hand rule, we can say that to the right of this current, the field will be into the page and to the left of this wire, the field will be out of the page. And we’ll put labels by each of those marks so we know which wire — wire 𝑥 — is creating the field in those directions.
Considering the other wire, wire 𝑦, the current in that wire will create a field that moves out of the page to the left of the wire and into the page to the right. Since we’re looking for the point on our diagram where the total magnetic field is zero, since in the region to the left of both wires which includes point 𝐴, the fields created by both wires are out of the page, that means they can’t possibly sum up to zero since they’re pointing in the same direction. So point 𝐴 cannot be a place where the magnetic field is zero.
In a similar way, since to the right of both of the wires, the fields created by each of them are into the page, once again, they can’t sum up to zero. So point 𝐸 can’t be a place where the magnetic field is zero. This leaves us with the three points in the middle region between the two wires, separated by the same distances as marked by the double hash marks.
In this region, we see that the direction of the field created by wire 𝑥 and the direction of the field created by wire 𝑦 oppose one another. If we were to pose it that the magnitude of the magnetic field created by wire 𝑥 is equal to that created by wire 𝑦, then, using our mathematical relationship, we can write that 𝜇 nought times one amp over two 𝜋 times 𝑟 sub 𝑥 is equal to 𝜇 nought times three amps over two 𝜋 times 𝑟 sub 𝑦.
In this equation, the factor of 𝜇 nought divided by two 𝜋 cancels out from both sides. And then cross multiplying to solve for 𝑟 sub 𝑦 divided by 𝑟 sub 𝑥, we find that this ratio is equal to three amps over one amp or simply three over one. This relationship tells us that the point on this diagram at which the magnetic field is zero is the point three times as far from wire 𝑦 as it is from wire 𝑥.
Looking at the middle portion of our diagram, we see that point 𝐵 meets this condition: the distance from 𝐵 to wire 𝑦 is three times the distance from 𝐵 to wire 𝑥. This means it’s at point 𝐵 that the magnitude of 𝐵 sub 𝑥 is equal to the magnitude of 𝐵 sub 𝑦. In other words, the magnetic field is zero.