Imagine placing a magnetic material inside the coil, as illustrated in Fig. 1. From practical experience, we know that the magnetic field will attract this material.

Figure 1: A magnetic material partially inserted inside the coil.

We will need a more sophisticated model to calculate an accurate value for the force, however, we can investigate it qualitatively with this toy model. Note that we labeled the vertical distance that is not filled by the core material as \(s\). The most practical way to compute \(\mathbf{B}\) is to use the concept of magnetomotor force, MMF, which is denoted by \(\mathcal{F}\). In the case of a coil \(\mathcal{F}=N I\), \(N\) is the number of turns and \(I\) is the current. If we express our fields in terms of magnetization field \(\mathbf{H}=\mu\mathbf{B}\), we can write the amperes law as follows: \[\begin{eqnarray} NI=H_c (H-s)+H_0 s =B\frac{H-s}{\mu_c} +B\frac{s }{\mu_0} \tag{1}. \end{eqnarray}\] Solving for \(B\), we get: \[\begin{eqnarray} B=\frac{\mu_r \mu_0 N I}{H}\frac{1}{1+\frac{s}{H}(\mu_r-1) }\tag{2}, \end{eqnarray}\] where \(\mu_r\equiv \mu_c/\mu_0\) is the relative permeability of the core material. The energy density of this setup will be a function of \(s\) and the vertical axis \(z\) since \(\mu\) changes. \[\begin{eqnarray} \mathcal{E}(s,z)=\frac{\mathbf {B}^2(s,z)}{2 \mu(z)} \tag{3}. \end{eqnarray}\] Integrating this over the volume we get \[\begin{eqnarray} E(s)&=&\pi R^2 \int d z \mathcal{E}(s,z) = \pi R^2 \int d z \frac{\mathbf {B}^2}{2 \mu(z) }=\pi R^2 \frac{\mathbf {B}^2}{2} \left[\frac{H-s}{ \mu_c}+\frac{s}{ \mu_0} \right] \nonumber\\ &=& \frac{ \pi R^2 \mu_r \mu_0 N^2 I^2}{2 H}\frac{1}{1+\frac{s}{H}(\mu_r-1) } \tag{4}. \end{eqnarray}\] The force on the material can be calculated simply by taking the derivative of the energy with respect to \(s\):

\[\begin{eqnarray} F(s)=\frac{dE(s)}{ds} &=& -\frac{ \pi R^2 \mu_r^2 \mu_0 N^2 I^2}{2 H^2}\frac{1}{\left(1+\frac{s}{H}(\mu_r-1)\right)^2 } \tag{5}. \end{eqnarray}\] Let’s divide the force by the volume, and re-write it in a slightly different form:

\[\begin{eqnarray} \frac{F(s)}{V}= -\frac{ \mu_r^2 \mu_0 N^2 I^2}{2 H^3}\frac{1}{\left(1+\frac{s}{H}(\mu_r-1)\right)^2 }\propto \frac{(\mu_r N I)^2}{ H^3} \tag{6}. \end{eqnarray}\] What this toy model teaches us is that, the acceleration of the core material, \(\propto \frac{F(s)}{V}\) can be maximized by maximizing this expression \(\frac{(\mu_r N I)^2}{ H^3}\). Note that, not all of the quantities can be varied independently. For example, to support higher currents, we will need thicker wires, which will reduce the number of turns we can fit in a given length \(L\). Furthermore, larger \(N\) will result in larger internal resistance for the coil, limiting the maximum current we can push. We will revisit this when we are modeling a more realistic solenoid.

So far we have ignored the fields outside the coil, but this doesn’t mean they are not important. We can use the MMF technique one more time to understand the effect of the external volume to modify Eq. (1) as follows \[\begin{eqnarray} NI=H_c (H-s)+H_0 s +H^{o}_{eff}L^{o}_{eff} =B\frac{H-s}{\mu_c} +B\frac{s }{\mu_0}+B\frac{L^{o}_{eff} }{\mu_o} \tag{7}, \end{eqnarray}\] where \(H^{o}_{eff}\) represents the average effective field outside the coils, \(L^{o}_{eff}\) is the average effective length the field travels, and finally \(\mu_o\) is the permeability of the material outside. Solving for \(B\), and going through the same steps, we can calculate the force on the core material as:

\[\begin{eqnarray} F(s) &=& -\frac{ \pi R^2 \mu_r^2 \mu_0 N^2 I^2}{2 H^2}\frac{1}{\left(1+\frac{s}{H}(\mu_r-1)+\frac{L^{o}_{eff}}{H} \frac{\mu_c}{\mu_o}\right)^2 } \tag{8}. \end{eqnarray}\] This equation tells us that the force doesn’t simply come from within, it is affected by the properties of the external volume. Qualitatively speaking, the force is determined by the flux we can push through, and that flux is inversely proportional to the reluctance of the whole set up. If we can make the flux travel with ease by reducing the external path to close the magnetic loop, we can increase \(B\). In order to do that, we have this knob: \(\frac{L^{o}_{eff}}{\mu_o}\). We can enclose the solenoid in a magnetic material such that this ratio is minimized.

It is also important to note that the \(s\) dependence of the force will be somewhat accurate only when the core material extends far out from one end, and is inserted well into the coil so that fringe fields are not significant. A more precise calculation requires the computation of fields outside the coil, and that is highly non trivial. However, we will soon argue that a good design should include shields around the coil to localize the external fields. This will also save us from dealing with complicated magnetic fields.

We have extracted a few nuggets of useful information from the toy model:

1. The acceleration of the core material can be adjusted by tuning this factor: \((\mu_r N I)^2/L^3\).
2. Building a return path for the flux increases the acceleration.
3. Eddy losses can be minimized by laminating the conductive material.

All of these observations are implemented in a typical solenoid, as illustrated in Fig. 2

Figure 2: Diagram of a realistic solenoid. Note that this particular set up is designed to pull the plunger inward. It can be modified easily to move the plunger outward. In both cases the modeling will be the same.