Two infinitely long straight conductors carrying current in the same direction attract each other

A pair of parallel wires serves to illustrate a principle that French scientist André-Marie Ampère was the first to comprehend, back in 1820.

The first person to discover evidence that electricity and magnetism are related phenomena was Hans Christian Ørsted. In the midst of a lecture on electricity, Ørsted noticed that a wire carrying current was able to deflect a compass needle. Unable to develop a plausible explanation, Ørsted published his findings in 1820. This news sent shockwaves through the scientific community and instigated numerous investigations into the matter. One of the scientists who immediately began to expand upon Ørsted’s work was Frenchman André-Marie Ampère. Soon after, Ampère found that the magnetic fields created by parallel current-carrying wires interact with one another, as demonstrated in this tutorial.

The direct current circuit in this tutorial includes two parallel straight wires (in red). These wires can be arranged in a series circuit, as they are when this tutorial first opens, or in a parallel circuit. In the series circuit arrangement, the parallel wires are linked by a connecting wire into a circuit that allows current along a single path. As a result, the current travels one way down one wire, and in the opposite direction down the second wire.

You can change this to a parallel circuit by clicking on the radio button; in this scenario, the current forks when it reaches the parallel wires; some current goes down one of the wires, the rest down the second. The current down both wires travels in the same direction.

Watch how the parallel wires behave in each of these set-ups. When either circuit type is selected, the Knife Switch can be lowered to complete the circuit by clicking either the switch itself or the blue Run button. Notice a stream of yellow electrons traveling through the circuit; flowing (as they always do) from negative to positive – opposite the direction of conventional current. To halt the flow of current, click on the red Stop button or the knife switch. Clicking the blue Pause button will let you examine the process in mid-stream.

As you can see, the wires in the series circuit repel one another, while the wires in the parallel circuit attract one another.

This is explained by the right hand rule, which helps visualize how a magnetic field (depicted by the blue field lines above) around a wire travels. Extend your thumb in the direction of the conventional current, then allow your fingers to curve: The magnetic field circling the wire (represented by your thumb) travels in the direction that your curved fingers are pointing.

So if you have two current-carrying, parallel wires with magnetic fields circling around them in the same direction, they will attract each other, as shown in the tutorial; at the point at which their respective magnetic fields intersect, they are traveling in opposite directions, and opposites attract.

Similarly, if you have two parallel wires with current traveling in opposite directions, as you do in the series circuit, then the magnetic fields of the two wires will be traveling in the same direction at the point at which they intersect, and therefore repel each other.

Ampère was able to mathematically describe this type of magnetic force between electric currents, formulating what is known as Ampère’s law.

Consider two infinite parallel straight wires, a distance \(h\) apart, carrying upwards currents, \(I_{1}\) and \(I_{2}\), respectively, as illustrated in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): Two parallel current-carrying wires will exert an attractive force on each other, if their currents are in the same direction.

The first wire will create a magnetic field, \(\vec B_{1}\), in the shape of circles concentric with the wire. At the position of the second wire, the magnetic field \(B_{1}\) is into the page, and has a magnitude:

\[\begin{aligned} B_{1}=\frac{\mu_{0}I_{1}}{2\pi h} \end{aligned}\]

Since the second wire carries a current, \(I_{2}\), upwards, it will experience a magnetic force, \(\vec F_{2}\), from the magnetic field, \(B_{1}\), that is towards the left (as illustrated in Figure \(\PageIndex{1}\) and determined from the right-hand rule). The magnetic force, \(\vec F_{2}\), exerted on a section of length, \(l\), on the second wire has a magnitude given by:

\[\begin{aligned} F_{2}=I_{2}||\vec l\times\vec B_{1}||=I_{2}lB_{1}\frac{\mu_{0}I_{2}I_{1}l}{2\pi h} \end{aligned}\]

where we used the fact that the angle between \(\vec l\) and \(\vec B\) is \(90^{◦}\). We expect, from Newton’s Third Law, that an equal and opposite force should be exerted on the first wire. Indeed, the second wire will create a magnetic field, \(\vec B_{2}\), that is out of the page at the location of the first wire, with magnitude:

\[\begin{aligned} B_{2}=\frac{\mu_{0}I_{2}}{2\pi h} \end{aligned}\]

This leads to a magnetic force, \(\vec F_{1}\), exerted on the first wire, that points to the right (from the right-hand rule). On a section of length, \(l\), of the first wire, the magnetic force from the magnetic field, \(\vec B_{2}\), has magnitude:

\[\begin{aligned} F_{1}=I_{1}||\vec l\times\vec B_{2}||=I_{1}lB_{2}\frac{\mu_{0}I_{1}I_{2}}{2\pi h} \end{aligned}\]

which does indeed have the same magnitude as the force exerted on the second wire. Thus, when two parallel wires carry current in the same direction, they exert equal and opposite attractive forces on each other.

Exercise \(\PageIndex{1}\)

Figure \(\PageIndex{2}\): Two wires that carry current in opposite directions.

Two parallel wires carry current in opposite directions, as shown in Figure \(\PageIndex{2}\). What force do they exert on each other?

There will be no force, since the currents cancel.
There will be an attractive force between the wires.
There will be a repulsive force between the wires.

Answer

The attractive force between two wires used to be the basis for defining the Ampere, the S.I. (base) unit for electric current. Before 2019, the Ampere was defined to be “that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in vacuum, would produce between these conductors a force equal to \(2 × 10^{−17}\text{N}\) per meter of length”. Recently, the definition was updated to be based on defining the Coulomb in such a way that the elementary charge has a numerical value of \(e = 1.602 176 634 × 10^{−19}\text{C}\), and the Ampere corresponds to one Coulomb per second.

The force between two wires is a good system to understand how any physical quantity cannot depend on our choice of the right-hand to define cross-products. As mentioned in the previous chapter, any physical quantity, such as the direction of the force exerted on a wire, will always depend on two successive uses of the right hand. In this system, we first used the right-hand rule for axial vectors to determine the direction of the magnetic field from one of the wires. We then used the right-hand rule to determine the direction of the cross-product to determine the direction of the force on the other wire. You can verify that you get the same answer if you, instead, use your left-hand to define the direction of the magnetic field (which will be in the opposite direction), and then again for the cross-product. This also highlights that the magnetic field (and the electric field) is just a mathematical tool that we use to, ultimately, describe the motion of charges or compass needles.

Exercise \(\PageIndex{2}\)

When current is flowing in a straight cable, how to you expect the charges to be distributed radially through the cross-section of the cable?