22.10 Magnetic Force between Two Parallel Conductors - College Physics for AP® Courses

Learning Objectives

By the end of this section, you will be able to:

Describe the effects of the magnetic force between two conductors.
Calculate the force between two parallel conductors.

The information presented in this section supports the following AP® learning objectives and science practices:

2.D.2.1 The student is able to create a verbal or visual representation of a magnetic field around a long straight wire or a pair of parallel wires. (S.P. 1.1)
3.C.3.1 The student is able to use right-hand rules to analyze a situation involving a current-carrying conductor and a moving electrically charged object to determine the direction of the magnetic force exerted on the charged object due to the magnetic field created by the current-carrying conductor. (S.P. 1.4)

You might expect that there are significant forces between current-carrying wires, since ordinary currents produce significant magnetic fields and these fields exert significant forces on ordinary currents. But you might not expect that the force between wires is used to define the ampere. It might also surprise you to learn that this force has something to do with why large circuit breakers burn up when they attempt to interrupt large currents.

The force between two long straight and parallel conductors separated by a distance $r$ can be found by applying what we have developed in preceding sections. Figure 22.44 shows the wires, their currents, the fields they create, and the subsequent forces they exert on one another. Let us consider the field produced by wire 1 and the force it exerts on wire 2 (call the force $F_{2}$ ). The field due to $I_{1}$ at a distance $r$ is given to be

B_{1} = \frac{μ_{0} I_{1}}{2 πr} .

22.30

Figure a shows two parallel wires, both with currents going up. The magnetic field lines of the first wire are shown as concentric circles centered on wire 1 and in a plane perpendicular to the wires. The magnetic field is in the counter clockwise direction as viewed from above. Figure b shows a view from above and shows the current-carrying wires as two dots. Around wire one is a circle that represents a magnetic field line due to that wire. The magnetic field passes directly through wire two. The magnetic field is in the counter clockwise direction. The force on wire two is to the left, toward wire one.

Figure 22.44 (a) The magnetic field produced by a long straight conductor is perpendicular to a parallel conductor, as indicated by RHR-2. (b) A view from above of the two wires shown in (a), with one magnetic field line shown for each wire. RHR-1 shows that the force between the parallel conductors is attractive when the currents are in the same direction. A similar analysis shows that the force is repulsive between currents in opposite directions.

This field is uniform along wire 2 and perpendicular to it, and so the force $F_{2}$ it exerts on wire 2 is given by $F = IlB sin θ$ with $sin θ = 1$ :

F_{2} = I_{2} {lB}_{1} .

22.31

By Newton’s third law, the forces on the wires are equal in magnitude, and so we just write $F$ for the magnitude of $F_{2}$ . (Note that $F_{1} = - F_{2}$ .) Since the wires are very long, it is convenient to think in terms of $F / l$ , the force per unit length. Substituting the expression for $B_{1}$ into the last equation and rearranging terms gives

\frac{F}{l} = \frac{μ_{0} I_{1} I_{2}}{2 πr} .

22.32

$F / l$ is the force per unit length between two parallel currents $I_{1}$ and $I_{2}$ separated by a distance $r$ . The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions.

Making Connections: Field Canceling

For two parallel wires, the fields will tend to cancel out in the area between the wires.

There are two small circles with dots in the center representing wires going in the same direction. Each circle has three progressively larger circles with arrows pointing in counter-clockwise positions representing the magnetic fields going in the same direction. The center circles are close enough that the first outer circle is between the two circles and the second outer circle bisects the other’s center circle.

Figure 22.45 Two parallel wires have currents pointing in the same direction, out of the page. The direction of the magnetic fields induced by each wire is shown.

Note that the magnetic influence of the wire on the left-hand side extends beyond the wire on the right-hand side. To the right of both wires, the total magnetic field is directed toward the top of the page and is the result of the sum of the fields of both wires. Obviously, the closer wire has a greater effect on the overall magnetic field, but the more distant wire also contributes. One wire cannot block the magnetic field of another wire any more than a massive stone floor beneath you can block the gravitational field of the Earth.

Parallel wires with currents in the same direction attract, as you can see if we isolate the magnetic field lines of wire 2 influencing the current in wire 1. Right-hand rule 1 tells us the direction of the resulting magnetic force.

The two small circles with dots representing wires are shown again in this diagram without the circles representing the magnetic fields. A blue arrow pointing down going through wire 1 is labeled Magnetic field due to wire 2. A red line from the center of wire 1 pointing to the right toward Wire 2 is labeled Magnetic force on wire 1 due to magnetic field of wire 2.

Figure 22.46 The same two wires are shown, but now only a part of the magnetic field due to wire 2 is shown in order to demonstrate how the magnetic force from wire 2 affects wire 1.

When the currents point in opposite directions as shown, the magnetic field in between the two wires is augmented. In the region outside of the two wires, along the horizontal line connecting the wires, the magnetic fields partially cancel.

There are two small circles with the one on the left having an X and the one on the right having a dot (representing opposite directions). Each circle has three progressively larger circles with the arrows on the left pointing clockwise and the arrows on the right pointing counter-clockwise. The center circles are close enough that the first outer circle is between the two circles and the second outer circle bisects the other’s center circle.

Figure 22.47 Two wires with parallel currents pointing in opposite directions are shown. The direction of the magnetic field due to each wire is indicated.

Parallel wires with currents in opposite directions repel, as you can see if we isolate the magnetic field lines of wire 2 influencing the current in wire 1. Right-hand rule 1 tells us the direction of the resulting magnetic force.

The two small circles with an x on the left and dot on the right representing wires with opposite currents are shown in this diagram. A blue arrow pointing down going through wire 1 is labeled Magnetic field due to wire 2. A red line from the center of wire 1 pointing to the left away from wire Wire 2 is labeled Magnetic force on wire 1 due to magnetic field of wire 2.

Figure 22.48 The same two wires with opposite currents are shown, but now only a part of the magnetic field due to wire 2 is shown in order to demonstrate how the magnetic force from wire 2 affects wire 1.

This force is responsible for the pinch effect in electric arcs and plasmas. The force exists whether the currents are in wires or not. In an electric arc, where currents are moving parallel to one another, there is an attraction that squeezes currents into a smaller tube. In large circuit breakers, like those used in neighborhood power distribution systems, the pinch effect can concentrate an arc between plates of a switch trying to break a large current, burn holes, and even ignite the equipment. Another example of the pinch effect is found in the solar plasma, where jets of ionized material, such as solar flares, are shaped by magnetic forces.

The operational definition of the ampere is based on the force between current-carrying wires. Note that for parallel wires separated by 1 meter with each carrying 1 ampere, the force per meter is

\frac{F}{l} = \frac{(4π \times 10^{- 7} T \cdot m/A) {(1 A)}^{2}}{(2 π) (1 m)} = 2 \times 10^{- 7} N/m.

22.33

Since $μ_{0}$ is exactly $4π \times 10^{- 7} T \cdot m/A$ by definition, and because $1 T = 1 N/ (A \cdot m)$ , the force per meter is exactly $2 \times 10^{- 7} N/m$ . This is the basis of the operational definition of the ampere.

The Ampere

The official definition of the ampere is:

One ampere of current through each of two parallel conductors of infinite length, separated by one meter in empty space free of other magnetic fields, causes a force of exactly $2 \times 10^{- 7} N/m$ on each conductor.

Infinite-length straight wires are impractical and so, in practice, a current balance is constructed with coils of wire separated by a few centimeters. Force is measured to determine current. This also provides us with a method for measuring the coulomb. We measure the charge that flows for a current of one ampere in one second. That is, $1 C = 1 A \cdot s$ . For both the ampere and the coulomb, the method of measuring force between conductors is the most accurate in practice.