A constant current flowing in an electromagnetic system, such as two circular coils[1], magnets and magnets[2], coils and magnets[3], or magnets and solenoids[4], is known to generate a magnetic force between them. The Biot–Savart law for magnetic fields was recognized after Coulomb reported that current flows through two wires when a magnetic force acts between them[5]. It was later discovered that Lorentz’s formula for magnetic forces combines the magnetic force law and the Biot–Savart law[5]. Thus, if the magnetic field generated by a conductor is B, the force dF acting on the small segment dl through which the current I flows is often obtained using the Lorentz magnetic force equation:[3,6,7]

Moreover, because the magnetic force dF→ is a conservative force, a method for obtaining this force using the gradient of the magnetic potential energy U(r) is proposed[8,9].

Furthermore, the concepts of virtual work and displacement have been introduced into electromagnetics and used to calculate the magnetic force and torque[5,10]. The magnetic force F between currents I and I′ flowing through two coils along the z-axis is calculated as the derivative of the mutual inductance M[1,4,11]:

Meanwhile, because the mutual inductance equation between the two coils is known, the magnetic torque N between the two current loops is mostly calculated using the derivative of the mutual inductance for the rotation angle θ[12]

The magnetic force and torque, which act on coils in a magnetic field, are sufficient to attract students’ attention. Thus, conducting experiments on them and comparing the experimental and theoretical values will be worthwhile. However, the magnetic force equations (Eqs. (1)–(3)) in the above three methods can be complex for college students because they exhibit different forms and likely originate from different physical concepts. Therefore, for students to understand, it is necessary to show that Eqs. (1)–(3) are the same expressions. Furthermore, the three equations can be used to express the magnetic torque N, they are all equal. Many college textbooks cover F and N, but they never explain the equivalence between the expressions. Only in a textbook is the equivalence of Eqs. (1) and (3) introduced without any arguments about the equivalence of N[5].

In this study, the magnetic force equations (Eqs. (1)–(3)) are shown to be equivalent. Additionally, the magnetic force and torque equations derived in a previous study[10,11] contain elliptic integrals of the first and second types, which are complicated. Thus, by assuming a “special case,” we attempted to prove their equivalence by obtaining an analytic function that college-level students can use. The analytical function comprised only the elementary functions of the magnetic force and torque. Here, the “special case” implies that “a system composed of a circular coil with a large diameter and a circular coil with a small diameter was coaxial, and one coil was inclined at a small angle.” This is an approximate condition used in this study. As argued in Sec. III, in such a special case, the available formulas for magnetic fields of coils are known within the approximate limitations without counting on the elliptic functions[13].

To determine the relationship of F and N in terms of η=(x,y,z,r⋯) and θ, using the generalized coordinate q, the virtual work δW can be expressed in terms of the virtual displacement δq[14]:

where Q is the generalized force, which is given by

and U is the potential energy. If q in Eq. (5) is a coordinate that determines the distance traveled in a certain direction, Eq. (5) is expressed as

where F is the force. Furthermore, if q is a coordinate that determines the rotation angle along a certain axis, Eq. (5) is expressed as

where N is the torque[14]. Furthermore, if q is set as η=(x,y,z,r⋯) in Eq. (7), it becomes

and in Eq. (6), it is written as

If q is set as θ (or ϕ) in Eq. (8), it becomes

and in Eq. (6), it is given by

Figure 1 shows a magnetic dipole in a nonuniform magnetic field B→. U of the magnetic dipole with a magnetic moment μ→ within B→ is expressed as

Figure 1. Magnetic dipole in a nonuniform magnetic field B→.

and substituting Eq. (13) in Eqs. (10) and (12) yields

Note that Eqs. (13) and (14) can be used for all magnetic dipoles with the magnetic moment μ→, irrespective of the size and shape of the current loops[9].

For I flowing in a planar coil of area A, μ→=I∫dAn^ (n^ is the unit vector of dA) and the magnetic flux Φ=∫ B→⋅dAn^. Thus,

is expressed using Eq. (14). Similarly, Eq. (15) is given by

If there are two coils and B→ is formed by the current I′ of one coil, the mutual inductance M of the other coil is defined as Φ=MI′. Thus, Eqs. (16) and (17) can be expressed as

Considering that the positions of two coils change by I and I′,resulting in only a change in M, while I and I′ flowing through the two coils remain constant after the change in positions, Eqs. (18) and (19) are given by

Thus, Eqs. (10) and (12) are equivalent to Eqs. (20) and (21), respectively. Additionally, when the current I is constant in Eqs. (16) and (17), both equations have the following respective forms:

Equations (22) and (23) are also beneficial because F and N are related to the derivation of only Φ; they indicate that the magnetic force and torque exist when there is a change in the η or θ direction of the magnetic flux Φ.

Thus, F and N acting on the magnetic dipole in a nonuniform magnetic field can be obtained from Eqs. (10), (20), and (22) and Eqs. (12), (21), and (23), respectively, indicating that all three equations are equivalent. In other words, the spatial changes in the η and θ directions of the magnetic potential, magnetic flux, and mutual inductance are all related to the magnetic force and torque. Therefore, the magnetic force or torque can be determined using the easiest of these equations according to the given magnetic field and shape of the magnetic dipole.

Meanwhile, the Lorentz force F→ acting on the conducting wire L through which the current I in the magnetic field B→ flows can be expressed using Eq. (1):

We use Eq. (22) to demonstrate its equivalence with Eq. (24). As shown in Fig. 2, when a magnetic dipole, such as a current coil, is placed in a magnetic field, the magnetic force and torque act simultaneously if the directions of B→ and μ→ differ.

Figure 2. Virtual displacement of the current coil Δη→ in a nonuniform magnetic field B→.

Assume that the magnetic force performed virtual work and caused virtual displacement Δq→. Because Δq→ is the generalized virtual displacement, q^ can be η^ or θ^. Considering the case where Δq→ is Δη→ (Fig. 2), the magnetic flux dΦ passing through the area Δη→×dl→ is

If we consider the magnetic flux Φ(t) passing through an area S at a time t and Φ(t+dt) passing through an area S′(= the area connecting two circular loops shown in Fig. 2, the hatched area belongs to S′) at a time t+dt. For the entire circuit L comprising the magnetic dipole, using Eq. (25), Φ can be expressed as[15]:

Substituting Eq. (26) in (22) yields

and

Thus, Eqs. (27) and (28) correspond to the following:

Thus, as observed from Eqs. (28) and (29), Eq. (22) is equal to the Lorentz force of Eq. (24).

Similarly, we attempt to derive

from Eq. (23).

As shown in Fig. 3, when the virtual angular displacement Δq→ is set as Δϕ→, the magnetic flux dΦ passing through the area, aΔϕ→×dl→, can be expressed as

Figure 3. Virtual displacement Δϕ→ related to the rotation of the current coil in a nonuniform magnetic field B→.

which becomes

Substituting Eq. (32) in Eq. (23) and using ϕ=π/2−θ yields

As observed from Eq. (33), Eqs. (23) and (30) are equivalent.

The equivalence of Eqs. (22) and (24) has also been proved by another method using the vector potential and Neumann’s equation[5], but the equivalence of Eqs. (23) and (30) was newly introduced in this study.

We demonstrated that, despite being equal, the relational expressions for the forces and torques acting on the magnetic dipole in a nonuniform magnetic field exhibit different expressions. For example, we introduced the equality of force F_z and torque Nθ acting on each current flowing through circular coils C₁ and C₂, which were derived from the magnetic potential energy, mutual inductance, and the Lorentz force.

For calculation simplicity, we considered the case where one coil (C₁) has a larger diameter than the other coil (C₂) and the angle θ by which one coil is inclined to the central axis z is small, while the distance z between the two coils is large. Here, the magnetic flux passing through C₂ due to the magnetic field generated by C₁ was governed by B_z rather than Bρ[13]. Moreover, the change in the radial direction of B_z was negligible and can be to B_z on the central axis. In other words, the change in the radial direction of the magnetic field and the deviation in both sides of the coil due to the size of C₂ were not considered. The effect on the size of C₂ will be introduced during the experiments in subsequent studies. As shown in Fig. 4, when C₂ is small and far from C1,Bz generated by I flowing through C₁ is expressed along the z-axis:

Figure 4. Two circular coils through which the current flows. The current I′ is located far from the current I, and it is inclined by a small angle θ with respect to the central axis z.

Note that B_z in Eq. (34) is not the magnetic field of the ideal magnetic dipole because it depends on the denominator. Because a magnetic dipole, such as the current loop, has a finite radius a, Eq. (34) yields a more precise B_z. Equation (34) can be used in Eq. (13) in this case.

The magnetic flux Φ of C₂ by B_z is expressed as

Using Eqs. (20) and (21), the force and torque along the central axis due to the mutual inductance effect are expressed as

and

When C₁ and C₂ are parallel, θ=0 and

On the other hand, if the force is calculated using the magnetic potential U of Eq. (13):

and differentiating with respect to z and θ, yields

If the two coils are parallel, we obtain

Thus, Eqs. (36) and (41) and Eqs. (37) and (42) are equivalent.

Finally, the magnetic force is calculated using the Lorentz force (the magnetic force law). In Fig. 4, when the current I′dl flows in the magnetic field B→ generated by I of C₁, the force dF→ acting on the current length element dl→ of C₂ is expressed as

Here, the net force acting in the same direction (l^×ρ^) for the entire coil is

and the net torque acting in the opposite directions (l^×z^) with respect to the rotation axis is

According to Eqs. (46) and (47), the magnetic field that generates the net force is the Bρ component by C₁ and the magnetic field that generates the net torque is the B_z component by C₁. Bρ(ρ,z) produced by C₁ near the central axis is approximately expressed as[14]

and if ρ in Eq. (48) is small at the a′ position of C₂, we obtain

Figure 5 shows the different forces acting on C₂, which are analyzed to obtain dF₁ and dF₂. Because the forces act on the entire C₂ and the unit vectors are l^⊥ρ^, dF→1 is expressed as

Figure 5. (Color online) Force acting on the circular coil C₂ through which the current flows.

As shown in the figure, the direction of dF→1 is perpendicular to both l^ and ρ^. As a result, if the coil rotates by θ, the direction of l^ also changes. Therefore, dF1=dFz only when θ=0, and dFz=0 when θ=π/2. Thus, we obtain

such that we modify Eq. (50) to

Conversely, dF→2 (Fig. 5) is quite challenging to obtain. If dl→≡dll^ and dl→ is expressed as dl→1 parallel to the rotation axis, while dl→2 perpendicular to the rotation axis, the following equation is obtained:

As shown in the figure, the second term in Eq. (53), i.e., the current length elements dl→2 and dl→2′ at the two points P and P′, is the opposite. Thus, the net magnetic forces generated by the current length elements cancel each other via the symmetry of the coil. Therefore, according to Eq. (53), the first term related to dl→1 and dl→1′ in the same direction contributes to the magnetic force.

Because the directions of l^1×z^ are ρ^ and l^1⊥z^, from Eq. (53)

Furthermore, in Fig. 5, dl₁ = dlsinα and dl=a′dα; thus, the equation can be modified as

The torque dNθ along the rotation axis is twice the product of the force dF⊥ (=dFρsinθ) perpendicular to the plane C₂ and the distance a′sinα from the current length element to the rotation axis. Therefore, we obtain

As a result, the net torque Nθ acting on the entire C₂ is

Therefore, Eqs. (52) and (41) & Eqs. (57) and (42) are equivalent, only the sign is different. If we insert the direction of dF→1 in Eq. (51) or the opposite in Fig. 5, we obtain the expressions with the same sign. Although the magnetic force F_z obtained using the magnetic potential or mutual inductance effect is related to ∂Bz/∂z, i.e., the gradient of the magnetic field, the magnetic force F_z using the magnetic force law is directly related to Bρ, which may seem strange.

However, when the magnetic force is generated by Bρ and the loop is moved, the loop is located where the magnetic field changes. Therefore, F_z moves the coil to the point where the magnetic field changes.

Because μ0 is small in all the equations for F_z and Nθ obtained so far, experimentally measuring F_z and Nθ is challenging, except if I and I′ are extremely large. In the F_z and Nθ measurement experiment, I and I′ must be enlarged so that one of the coils can use a magnet or solenoid. When using a magnet or solenoid, the formula of the magnetic field in F_z and Nθ must be changed via integration of Eqs. (34) and (48). The results of F_z and Nθ calculated by changing the equation of the magnetic field and considering the magnetic flux based on the coil size, as well as experimentally measured F_z and Nθ are reported subsequently.

Three equations derived from the Lorentz force, the magnetic potential energy, and the mutual inductance effect of the magnetic force and torque acting between two magnetic dipoles were proven to be equivalent. Additionally, the magnetic force and torque acting between the circular coils of varying sizes through which constant current flows are equivalent, as determined by the changes in the magnetic potential energy, mutual inductance, and magnetic flux. The results of this study can be used to calculate the magnetic force and torque acting between other electromagnetic systems, such as coils and magnets, magnets and magnets, or magnets and solenoids, in addition to different circular coils. The results are expected to aid college-level students in understanding magnetic force and torque between two magnetic dipoles without conceptual confusion.