Is Feynman wrong about the principle of virtual work to find forces in a dielectric?

Question

In The Feynman Lectures on Physics, Vol II, 10–5 Fields and forces with dielectrics describes a method for finding the force between two charged conductors in a dielectric. I accept the first part of his description of its application:

Let us now ask what the force would be between two charged conductors in a dielectric. We consider a liquid dielectric that is homogeneous everywhere. We have seen earlier that one way to obtain the force is to differentiate the energy with respect to the appropriate distance. If the conductors have equal and opposite charges, the energy $U=Q^{2}/2C,$ where $C$ is their capacitance. Using the principle of virtual work, any component is given by a differentiation; for example,

$$F_{x}=-\frac{\partial U}{\partial x}=-\frac{Q^{2}}{2}\frac{\partial}{\partial x}\left(\frac{1}{C}\right).$$

Since the dielectric increases the capacity by a factor $\kappa$, all forces will be reduced by this same factor.

However; the following is contrary to my understanding of the application of the principle of virtual work:

One point should be emphasized. What we have said is true only if the dielectric is a liquid. Any motion of conductors that are embedded in a solid dielectric changes the mechanical stress conditions of the dielectric and alters its electrical properties, as well as causing some mechanical energy change in the dielectric. Moving the conductors in a liquid does not change the liquid. The liquid moves to a new place but its electrical characteristics are not changed.

The principle of virtual work, as I understand it, says we can make any imaginary change to a parameter to find the amount by which dependent variables change as a result. Suppose we only have a computer model of the configuration for which we want the forces. We can vary all the design variables so that there is no mechanical stress produced.

That is, if we want to change the separation between the plates of a capacitor, for example, we change the amount of dielectric between the plates at the same time we change $d$ by $\delta{d}$. First we calculate the energy of the original configuration. Then we modify the design by a small amount and recalculate the energy. Dividing the change in energy by the change $\delta{d}$ gives us an approximation of the derivative.

Does anybody agree that Feynman's asserted limitation to the application of virtual work in this circumstance is incorrect?

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Edit to add:

I don't have Lanczos's book with me, but I believe he gives an even clearer exposition of what I'm talking about than does Wells.

From Lagrangian Dynamics, by D. A. Wells, page 30, Real and Virtual Displacements: Virtual work. In discussing the behavior of a particle $m$ moving on a flat surface, Wells states:

$m$ moves along some definite path (determined by Newton's laws) in space and at the same time traces a line on the surface. During any given interval of time $dt,$ $m$ undergoes a specific displacement $ds (dx,dy,dz)$ measured say relative to stationary axes. Here $ds$ is referred to as an "actual" or "real" displacement.

Consider now any arbitrary displacement $\delta{s} (\delta{x},\delta{y},\delta{z})$ not necessarily along the above mentioned path. In this case $\delta{s}$ is referred to as a virtual displacement. For convenience in what follows, we mention three classes of such displacements: (a) $\delta{s}$ in any direction in space, completely disregarding the surface ...

For a virtual displacement of any type the "virtual work" done by $F$ is given by

$$\delta{W}=F\delta{s} \cos{F,\delta{s}}=F_x\delta{x}+F_y\delta{y}+F_z\delta{z}$$

Answer 1

Yes. Feynman is wrong about the applicability of the principle of virtual work to his example of a solid capacitor. To illustrate this, consider Exercise 2.1 from Exercises for the Feynman Lectures on Physics:

Use the principle of virtual work to establish the formula for an unequal-arm balance: $w_{1}l_{1}=w_{2}l_{2}$ (neglect the weight of the cross-beam).

Solution: Write an equation for the system's potential energy relative to some arbitrarily designated fixed height as

$$U=g\left(y_{1}m_{1}+y_{2}m_{2}\right).$$

In this application of the principle of virtual work changes in the parameters characterizing the system must not alter the potential energy.

Represent the potential energy at some fixed angle of the beam relative to the horizontal as a function of the lengths of the arms by

$$U_{\theta}\left[l_{1},l_{2}\right]=g\sin\left[\theta\right]\left(-l_{1}m_{1}+l_{2}m_{2}\right).$$

We could find the desired result by examining the consequence of altering $\theta.$ Instead, we shall examine the consequence of virtual displacements of the design parameters. The variation is

$$\delta U_{\theta}=\frac{\partial U_{\theta}}{\partial l_{1}}\delta l_{1}+\frac{\partial U_{\theta}}{\partial l_{2}}\delta l_{2}$$

$$=g\sin\left[\theta\right]\left(m_{2}\delta l_{2}-m_{1}\delta l_{1}\right).$$

Since we require $\delta U_{\theta}=0$ the result is

$$m_{2}\delta l_{2}=m_{1}\delta l_{1}.$$

Because this relation is independent of the length $l=l_{1}+l_{2}$ of the beam, we may replace the variations with finite displacements. Doing so and multiplying both sides by $g$ provides the advertised result.

The relevance of this example to the issue of Feynman's statement quoted in the original post is that the virtual change in the hypothetical configuration of the system did not entail any application of additional force in order to distort a particular physical implementation of the design. We are differentiating with respect to design parameters; in a “multi-verse” of all possible designs.

That Feynman had this wrong is stunning to me. I have successfully applied the method illustrated above to many of the exercises for the Feynman Lectures. I know it works.

Answer 2

The principle of virtual work, as I understand it, says we can make any imaginary change to a parameter to find the amount by which dependent variables change as a result.

While I am not sure, I think here lies the issue.

You cannot make any imaginary change to a parameter, only changes that can be made while time is frozen. If time was frozen and the dieletric was a liquid, you would be able to move the conductors closer together ("the liquid moves to a new place" and no properties are changed). If it was a solid, this virtual displacement would not be allowed, because you would change the conductor's properties.

In summary, I think this has to do with the fact that not all virtual displacements are allowed, but I must admit this is a subtlety I'm not entirely sure I understand. This is discussed in The Lazy Universe, by Jennifer Coopersmith:

A surprisingly tricky example is the case of a sliding block which is pushed across a table-top by a force, say, pushed by your finger (we ignore friction). The displacement of the block is anywhere on the surface whereas the reaction-force acts at right-angles to this surface preventing the block from burrowing down into the table. So far, this makes sense. But, hang on, there is also a reaction against your finger, from the block, and this reaction is in line with the block’s displacement. The trick is to appreciate that the block’s displacement due to the finger-push is an actual, not a virtual, displacement. We can hypothetically freeze the block (switch to a different reference frame) and get rid of the distraction of its actual motion. Then we realize that the finger can’t depress the block as if it were so much sponge-cake, as there is a reaction-force of the block against the finger. However, the finger is still allowed, infinitesimally, virtually, to move within the back face of the block, at right-angles to this reaction-force. This is a general result: for any virtual displacement, being ‘harmonious’ is the same thing as being in a direction perpendicular to the reaction forces.

Answer 3

No, Feynman is not wrong. He is showing an elegant way of relating the electrical force between the plates to the electrical energy stored. The concept is easiest to understand with the plates in empty space - although it would also work in a fluid like air. If the imaginary capacitor had a solid (like rubber) between the plates, then squeezing the plates together would result in a (large) mechanical force as well as an electrical one. He isn't trying to model that.