A New Perspective of the Force on a Magnetizable Rod inside a Solenoid

We analyze the familiar effect of the pulling of a magnetizable rod by a magnetic field inside a solenoid. We find that the analogy with the pulling of a dielectric slab by a charged capacitor is not as direct as usually thought. Indeed, there are two possibilities to pursue the analogy, according to the correspondence used, either → E B and → D H , or → E H and → D B . One of these results in an incorrect sign in the force, while the other gives the correct result. We avoid this ambiguity in the usual energy method applying a momentum balance equation derived from Maxwell’s equations. This method permits the calculation of the force with a volume integration of a force density, or with a surface integration of a stress tensor. An interpretation of our results establishes that the force acts at the interface and has its origin in Maxwell ́s magnetic stresses at the medium-vacuum interface. This approach provides new insights and a new perspective of the origin of this force.


Introduction
It is a well-known effect the pulling of a magnetizable rod into a solenoid where a magnetic field is established. There are many practical applications of this effect, like the Bendix mechanism, relays, etc., but how it arises is hardly discussed.

Journal of Electromagnetic Analysis and Applications
This effect is considered analogous to the pulling of a dielectric slab into a parallel plate capacitor. The similarity of these effects has been discussed by Boyer [1], and is also discussed in some intermediate texts on electromagnetism [2], [3], [4]. There is, however, a difference about the supposed origin of these effects in the electric and magnetic cases. In the electric case the force acting on the dielectric slab is usually assumed to be caused by the non-uniform fringing field outside the capacitor, [5], [6], [7], [8], [9], while in the magnetic case the fringing effects are explicitly neglected [2], [3], and there is no explanation of its origin.
In the electric case, therefore, the force is explained as the action of the non-uniform fringing field on the dipoles of the dielectric slab. We have shown [10] that this force arises rather from the uniform field inside the capacitor transmitted through the Maxwell stresses. In the magnetic case we have a uniform magnetic field inside the solenoid, which can only align the magnetic dipoles. Therefore, we have the question of how this uniform field can exert a net force on the magnetic dipoles. This is the question we address in the present work. It is rather a conceptual question that we will answer applying the Maxwellian notion of electromagnetic stresses. We use a particular electromagnetic momentum balance equation derived elsewhere [11] from Maxwell's equations.
We revise first (Section 2) the usual derivation of the magnetic force with energy methods. We find that the usual shortcut of copying the electric case by changing the fields → E B and → D H (1) leads to a wrong sign in the magnetic force. We also discuss the arguments of Landau and Lifshits [12] (Section 3) to show that sometimes the magnetic fields analogous to E and D are rather H and B , respectively. Our results point to the tension part of the magnetic stress tensor acting at the interface as the origin of this force (Section 4), explaining in this way how a uniform magnetic field can exert a net force on magnetizable matter. This constitutes a novel point of view that provides insights respect to the electromagnetic force, which is transmitted through stresses, as Faraday and Maxwell anticipated.
In Section 5 we present a new force density, Equation (61), from which the magnetic force can be obtained. In Sections 6 and 7 we give theoretical support to this force density showing how it appears in a momentum balance equation.
Our approach represents therefore a novel point of view that provides insights respect how the magnetic force arises.

Force on a Magnetic Rod inside a Solenoid with Constant Current I
The device is a solenoid of length L, with n turns per unit length, cross area 0 A and a constant current I circulating through it. A rod of magnetic material of permeability r µ and magnetic susceptibility m χ is partially introduced into the solenoid. A force appears that pulls the rod into the solenoid (Figure 1). Journal of Electromagnetic Analysis and Applications Usually this force is derived from the equation where the sub-index I indicates that the process considered occurs with constant current I, hence the positive sign in this equation, and the magnetic energy m U is given by (we follow Griffiths [9] and Purcell and Morin [13] and take the field Β as the magnetic field and H as an auxiliary field). Then, by expressing the H field in terms of the current, one can find the known result for the force on the magnetic rod calculating the change in energy in displacing the rod a distance z ∆ [2], [3], [4]. This force is With this procedure it is irrelevant the fringing magnetic field outside the solenoid and what happens at the interface, as noted by Wangsness [2] and Reitz et al. [3]. Now, when a magnetizable material is introduced into a magnetic field, the fields Η and B change, and therefore also the energy changes. If initially there is a vacuum inside the solenoid, the fields are 0 B and This expression can be transformed in order to have a volume integral only over the volume V' occupied by the material. In the electrostatic case there is also a transformation from, [2], [14], [15] ( ) from which it is deduced that In order to treat the magnetic case, Jackson [15] and Wangsness [2] propose the correspondence (12) in the electric case as expressed in Equation (10), obtaining Following the steps that in the electric case lead from Equation (10) to Equation (11), they obtain int mat mag Jac-Wan 0 equivalent to int mat mag Jac-Wan 0 Jackson [15] and Wangsness [2] call the attention to the difference in sign with respect to the electric case as given in Equation (11). This difference in sign implies a wrong direction in the force. Wangsness [2] obtains the correct result because he does not use result Equation (15) to calculate the force.
It is worth noting that the correct result is obtained if in the electrostatic case, In order to see the consequences of this difference in sign of the change in energy it is convenient to calculate the force with (2) and (15). The equation for the magnetization is (17) where Θ is Heaviside´s distribution. Then, from Equation (15), the constitutive relation (5), (6), and the fact that the H field is constant inside the solenoid, even inside the magnetic rod, one obtains for the change in energy ( ) 2 int mat mag Jac-Wan mat

of Electromagnetic Analysis and Applications
When the gradient is taken using After integration the final expression of the force is that is, the negative of the correct result (7).
Therefore, the correct result ensues if in the electrostatic result (10) the subs- is made. This points to the fact that in some cases the magnetic fields analogous to the electric fields are while in others it is the correspondence Equation (22).
We have in this respect Stratton's [16] observation: "Whatever the analytical advantages of the electrostatic analogy may be, it is well to remember that the physical structure of a field due to stationary distribu-

Landau and Lifshitz Thermodynamic Analysis
It is convenient to review with some detail the procedure followed by Landau and Lifshitz [12] with which it is established that in this case the correct analogy is (22).
In the electric case they begin with the work necessary to increase the charge of the system by q δ , This is the mechanical work performed by the electric field in bringing the charge q δ from infinite to its final position. In terms of the electric fields this work is Journal of Electromagnetic Analysis and Applications Then the change in Helmholtz's free energy, F U TS = − , is Note the correspondence Now, it is convenient to introduce a thermodynamic potential in which it appears δ E instead of δ D . This is done by means of a Legendre transformation * * .
The change of this free energy is Therefore, the expressions for the infinitesimal changes in the quantities expressed in terms of the charges and potentials of the conductors are, respectively, It is important to note the sign in Equation (32). We have then, besides Equa- In the magnetic case we have an analogous situation, but some care is necessary. We have instead of Equation (31) and Equation (32) the equations and L & L point out that in macroscopic electrodynamics the role analogous to the currents is played by the potentials, not by the charges.
These authors [12] observe that "It is useful to note that in macroscopic electrodynamics the currents (sources of the magnetic field) are mathematical analogues of potentials, not of charges

Where Does the Force Acts?
The energy methods for calculating the force exerted by a uniform magnetic field inside a solenoid on a magnetizable rod, do not permit establish where and how the force acts. Also, in the analogous electric case of the force exerted on a dielectric slab partially inserted into a parallel plate charged capacitor, the energy method does not allow to establish where and how the force acts. In the electric case, it is said [5], [6], [7], [8], and [9] that it is the non-uniform electrostatic fringing field acting on the dipoles of the dielectric what causes this force. In the magnetic case the fringing non-uniform magnetic field is explicitly neglected.
In order to establish clearly where and how the force arises, it is convenient to apply Maxwell's magnetic stress tensor. The force in terms of this tensor is, [17] d , where T  is the magnetic stress tensor.
( ) and σ is a closed surface. The relevant surfaces for the calculation are flat surface around the interfaces formed by a parallels surfaces close to the interface, one inside the rod and other in vacuum.
Since for linear magnetic media the constitutive relation (5) implies that B and H are parallel and in the k direction, the stress tensor becomes 1ˆ, where the unit dyad I  is ˆˆˆˆˆˆ.
For the relevant surface we have the differential surface elements (Figure 2) We have also that Then Equation (44) becomes, using Equation (43), Now we take into account the boundary condition which can be rewritten as This is the correct known result, Equation (7).
We can then conclude that the force acts at the interface magnetic rod-vacuum inside the solenoid and arises from the Maxwell magnetic stress tensor. This "magnetic pressure" was introduced and discussed by Maxwell, thus confirming that the fringing field is irrelevant.

The Force Density
As in the electrostatic case, usually we do not deal directly with a force density; this is obtained as a gradient of an energy density and in this way the result Equation (7) is obtained. Now, what is the force density in terms of the fields B or H and the magnetization M ?
What we want to explore is the possibility of a force density of type Journal of Electromagnetic Analysis and Applications ( ) analogous to the electric force density [15] ( ) from which the known result for the capacitor follows.
In this case the magnetization M depends only on z and is discontinuous at the interface. Since M is in the direction of H , which points in the z direction, we have and therefore new 0, showing that we have here a different situation from that in the electrostatic case. However, if we consider the magnetic energy density analogous to that of the electrostatic field, and calculate With the vector identity for the gradient of a scalar product, the force density which is the known result, [2], [3], [4], Equation (7), implying that the adequate force density is We have then, as in the electrostatic case, a new force density that gives the correct result. The electrostatic and magnetostatic cases seem very similar, but the force densities differ in both cases. This motivates looking for a more general momentum balance equation that includes, in the magnetic case, the force density Equation (61). In the next section we search for such balance equation.

Momentum Balance Equation for Magnetic Materials
The question is if the momentum balance equation [17], [18]

Conclusions
We have calculated, in a novel and insightful way that allows a physical explanation, the well-known force that arises when a magnetizable rod is partially introduced into the magnetic field of a solenoid. Our approach is an alternative to the usual calculation with the gradient of an energy. The usual calculation does not give any insight about where the force acts and how it arises, while our method does give such insights. This novel calculation is based on the volume integration of a force density and the surface integration of a magnetic stress tensor.
Though the force density may seem unfamiliar, it is part of a momentum balance equation derived directly from the macroscopic Maxwell equations. Indeed, these balance equations contain many force densities, for example the Helmholtz force density [11], with which this magnetic force can also be calculated. This method leads to results that indicate that this force is exerted at the rod-vacuum interface, where the magnetic field is uniform, and has its origin in Maxwell's magnetic stresses.
We also analyze the analogy with the electric case, used sometimes to calculate the force as a gradient of a magnetic energy. The proposed analogy consists in substituting in the electric case E with B and D with H . The analysis of this familiar effect shows that the classical theory of electromagnetism still contains interesting conceptual aspects that deserve attention.