Today's disquisition concerns some limitations on dimensional analysis and scaling arguments as applied to Child's law (or the Child-Langmuir Law, as it is sometimes called). For those who forgot, Child's law is an expression for the space-charge limited current through a vacuum tube diode in terms of the forward voltage across its electrodes. In general it says that the current is proportional to the 3/2 power of the voltage drop from the anode to the cathode, so it illustrates a gross violation of Ohm's law. A big reason for the discrepancy is that Ohm's law is a bulk effect in a conductor where the energy dissipation is spread out throughout the conductor and it is caused by local scattering of the charge carriers off of obstructions (dynamic lattice distortions, i.e. phonons) they have to navigate their way among. But the charge carriers in a vacuum diode continuously accelerate in the region between the electrodes in response to the electric field there and only dissipate the kinetic energy they carry when they are stopped by the anode plate and slowly leak off of it through its connection to the external rest of circuit.
If you must visualize the energy flow in the vacuum diode in terms of Poynting's vector (not very helpful for the purposes of this post) the Poynting energy flux, S, enters each part of the charge carrier beam from the side and that EM energy it carries is used to increase the kinetic energies of the charge carriers as they accelerate toward the anode. At the anode plate the electric field abruptly goes to (nearly) zero inside and there is a much more negligible Poynting flux entering it from the outside. Instead the carriers give up just about all their KE (and work function PE increase) to the anode plate's lattice thermal vibrational modes as they crash into it. But for the purposes of this post I think it is simplest to just imagine the charge carriers in the intervening vacuum as subject to a static electric potential energy field due to the static voltage gradient there. (But technically, for electron carriers, their KE inside the metal anode goes to whatever the KE is for the Fermi level in the conduction band happens to be, but I don't want to discuss that here and rather just consider the Fermi level/chemical potential inside as effectively defining the zero level of energy there, which suffices for our purposes here and for the conduction electron transport process inside the conducting metal.)
The great majority of the undergraduate E&M textbooks that consider Child's law at all assume (following Child's original derivation) a geometry for the vacuum diode which is essentially a flat parallel plate capacitor such that one conductor is heated to incandescence and acts as the cathode. The derivation assumes the distance, d, between the cathode and anode plates is tiny compared to the smallest lateral dimension of the conducting plates, so that d^2/A is a tiny number where A is the area of a plate. This is to make the fringing edge effects negligible. Unfortunately, essentially nobody builds vacuum tube diodes like this. Rather, they tend to use a cylindrical geometry with the much smaller cathode on the inside and a much larger anode on the outside. I suppose this makes heating a smallish cathode or using a direct filamentary cathode is easier to get the temperature up to incandescence as such a small electrode would not radiate away much of its heat very fast. In contrast the anode, being much larger, would more easily radiate whatever heat dissipated in it from the electron current crashing into it. This would tend to keep its temperature low enough for it to not have any thermionic emission of its own (so the diode would actually properly rectify an AC signal). The cross section shape of the cylinder varies in that rectangular, circular and ovoid shapes have been and are used. But a circular cross section shape for the cylinder is a fairly frequent/common shape. At one time I thought I saw the corresponding Child law derivation in a textbook for a somewhat more realistic cylindrical geometry with the cathode being a thin central wire coaxial with the cylindrical anode plate and with a low enough radius-to-length ratio for negligible fringing effects from the cylinder ends. But I can't seem to find any such a textbook among my stuff. Maybe I imagined it. Nevertheless it is almost as straightforward to solve for the forward biased I vs V function for the latter cylindrical geometry as for the former flat parallel plate geometry, and the results are nearly identical. In fact, the similarity of the two results is a main point of this lengthy post.
In the usual close-in parallel plate case the relationship between the current, I, and the forward voltage, V, across the electrodes has an exact solution of
Here ε_0 is the vacuum permitivity, A is the plate area, e is the magnitude of the electron charge, m is the electron mass, & d is the distance between the electrodes. Most of this formula could be deduced from a judicious use of dimensional analysis combined with some reasonable scaling arguments. Of course the 4/9 coefficient out front can't be found from such an argument. But pretty much the rest of the formula is determined from scaling and dimensional considerations. For example we know the current density must be constant across the region between the plates, and this makes I proportional to A. We also know the current density is the product of the space charge density and the electron velocity. The charge density enters the RHS of the governing Poisson equation with a factor of 1/ε_0 multiplying it. This means any solution for that charge density needs to have a factor of ε_0 multiplying it to get rid of the 1/ε_0 factor originally in the Poisson eqn. Also, since the current density is the charge density times the speed a factor of the speed at the anode need to be included in the formula for I. The static field that the electrons find themselves in is conservative, and they start out from the cathode at zero speed. (Essentially all of the initial thermal energy in the thermionically evaporated space charge is used up in getting only the most energetic electrons past the work function barrier and out into the vacuum, which has a very low probability since kT is tiny compared to any typical work function.) Now the gained KE of the electrons in traversing the region between the electrodes is just their lost potential energy from the voltage change across them we see that (m/2)*v^2 = e*V or v = sqrt(2*e*V/m), and so a factor of sqrt(2*e*V/m) also needs to appear in the formula for I. Now an extra factor of V will give our scaling dimensional analysis expression the units of A•m^2. So we need to divide by something with dimensions of m^2. The only relevant dimensional length left in the problem is the distance, d, between the plates. We expect that increasing d must lower the current because that weakens the field pulling on the charges. Thus we need to divide by d to whatever power is needed for the units to come out right. Clearly this power must be 2. So from purely a dimensional and scaling argument we have the gist of Child's law with only a dimensionless numerical factor left undetermined. Now this parallel plate model with a tiny inter-electrode gap is not very realistic for a real vacuum diode, and even if we stick with the flat plate geometry we expect the Child's law formula to still work with only a geometry dependent dimensionless factor out front left to be found. This factor would depend on both the d^2/A ratio and on the specific plate shape as we expect that a change in plate shape would affect the fringing field and thus its contribution to the correction/fudge factor.
In this flat parallel plate situation the space charge is at rest at the cathode, and is moving ever faster as the anode is approached, at which point the charges hit the anode at a top speed determined by their charge & mass and the forward voltage across the electrodes. Since the current density is the product of the charge density and the velocity, and since it is uniform in steady state between the flat plates the charge density is lowest just outside the anode and divergently great at the cathode. The very high space charge density near the cathode effectively screens the electric field there so the normal E-field right at the cathode is zero. If z (≤ d)is the distance from the cathode plate to some intermediate point between the plates then the potential at z is proportional to z^(4/3), the electric field is proportional to z^(1/3), the space charge density is proportional to z^(-2/3), the electron speed is proportional to z^(2/3), and the current density is constant independent of z.
Now, instead, consider the result for a *circular cylindrical geometry* for the anode electrode and a coaxial 1-dimensional linear cathode on the inside. In this case the exact solution for the current vs voltage relationship is
Here L is the length of the cylinder and b is the radius of the outer cylindrical anode electrode. In this case the tapering of the space charge density between the electrodes is much more severe than for the flat parallel plate case. But even so it is, nevertheless, still insufficient to effectively shield the electric field at the cathode. In fact the electric field is divergent there if the cathode radius is really zero (but this is not possible in practice, anyway). In particular, if r (≤ b) is the radial distance from the infinitesimally thin central cathode wire to some intermediate point between the cathode and anode then the potential at r is proportional to r^(2/3), the electric field is proportional to r^(-1/3), the space charge density is proportional to r^(-4/3), the electron speed is proportional to r^(1/3), and the current density is proportional to r^(-1). So we see the behavior appears to be drastically different between these two geometrically exactly solvable idealized cases.
But, remarkably, the Child's law formula for the thin cathode cylindrical geometry actually has exactly the same form as the previously considered flat plate geometry. This can be seen by noticing that the area, A, of the cylindrical anode electrode is 2*π*L*b. If we write the cylindrical formula in terms of A instead of the cylinder length, L, we get
Recall that in this case the anode radius, b, is also the distance between the cathode and anode (because the very thin cathode is assumed to have a zero radius). This makes the cylindrical formula and the flat plate formula have the very same form as long as we interpret A only to be the collection area of the anode and don't relate it to the cathode's area. I had noticed this coincidence of formulas a number of years ago and was puzzled by it considering how different these 2 cases are in their distributions of charge, current, potential, fields, etc. Is this coincidence of these 2 Child's law formulas just an accident, or is there some deeper identity between them which enforces the same resulting formula? Recently, i.e. since retiring from full time teaching, I've had some spare time to look into the matter. I report here what I found out about it.
To help answer the question about the coincidence of the above two highly idealized exact solutions we'll investigate a more realistic (but, unfortunately, no longer exactly solvable) slight generalization of the cylindrical geometry case. We know that *both* the close-in parallel plate geometry *and* the cylindrical geometry with an infinitely thin concentric wire cathode are not realistic models even if they both happen to be exactly solvable. One thing we can do to make the cylindrical geometry more realistic is to relax the requirement that the cathode be a length of very thin wire. So our new slightly generalized model still has full cylindrical symmetry, but the cathode now is also a circular cylinder coaxial with the anode cylinder but of a smaller radius, a. Thus a is the radius of the cathode cylinder and b is the radius of the anode cylinder (where 0< a < b). The next thing to notice is that even though this generalized model is no longer exactly solvable in closed form it has a limiting case that *includes* the parallel plate geometry. Notice that when a is so close to b that the b - a distance between the electrodes is infinitesimal compared to the radius a then we effectively have the parallel plate case in that a approaching b limit. Next let's imagine that the reason the same formula worked for both the a -> 0 and a-> b limits is because it actually works for all a/b ratios, including those opposite limiting cases. IOW, we consider a hypothesis that the Child's law formula for any a/b ratio has the structure
I =(4/9)*ε_0*A*sqrt(2*e/m)*(V^(3/2))/(b - a)^2
where A is the area of the outer anode electrode and b - a is the radial distance between the electrodes. Now we realize that having two opposite limiting cases have the same value is certainly no guarantee that all the intervening cases also have the same value since 2 matching endpoints on a curve doesn't mean the curve's function is a constant. But it does open up the possibility that that may be the case. We just need to investigate the intermediate region of the curve between those matching endpoints to see if the hypothesis holds up, or not. To allow for the possibility that the hypothesis is false we modify our Child's law formula to include a correction/fudge factor function that depends explicitly on the a/b ratio. If the fudge factor function is always unity independent of the a/b ratio then the hypothesis is confirmed. If the fudge factor function ends up being a complicated function that only has the same unity value when a/b → 0 and when a/b → 1, then the hypothesis is falsified. But, whatever the fudge factor function is, its value as a function of a/b tells us the answer for the Child's law formula for any coaxial cylindrical geometry (with negligible fringing effects near the cylinders' ends). Our Child's law I vs V formula then becomes
I =(4/9)*K(a/b)*ε_0*A*sqrt(2*e/m)*(V^(3/2))/(b - a)^2
Here the dimensionless fudge factor function is K(x) (whose limiting values are K(0) = 1 and K(1) = 1). I've numerically solved for the K(x) function for 0 < x < 1 and also made a power series expansion about the a → b close-in plate limit matching the expansion coefficients through the first 5 nonzero order terms in powers of (b/a - 1). I then converted the power series to the simplest rational function whose power series expansion matches the exactly calculated series terms, and whose value in the a → 0 limit is 1. The result is the K(x) function is *not* the unity constant, and so the hypothesis that K(x) might be always 1 is falsified. Actually, it appears that the fact that K(x) has K(0) = K(1) is, indeed, an accidental coincidence because there is no underlying symmetry that forces K(x) to be constant. For anyone who may be interested, the rational function approximation for K(x) is
Here are some particular values comparing the numerically calculated exact solution (within round-off error using a 4th order Runge-Kutta DE algorithm with a very small step size on the relevant dimensionless DE) with the above rational function approximation for K(x):
a/b = 32/33, K(32/33)_exact = 0.9939195285, K(32/33)_approx = 0.9939209702, error = 0.00014505%
a/b = 16/17, K(16/17)_exact = 0.9881646706, K(16/17)_approx = 0.9881656822, error = 0.00010237%
a/b = 4/5, K(4/5)_exact = 0.9591883773, K(4/5)_approx = 0.9591886487, error = 0.000028295%
a/b = 2/3, K(2/3)_exact = 0.9310932688, K(2/3)_approx = 0.9310904791, error = -0.00029962%
a/b = 1/2, K(1/2)_exact = 0.8951999485, K(1/2)_approx = 0.8951655449, error = -0.0038387%
a/b = 1/4, K(1/4)_exact = 0.8432160510, K(1/4)_approx = 0.8426996451, error = -0.061242%
a/b = 0.1183777, K(0.1183777)_exact = 0.8273988891, K(0.1183777)_approx = 0.8260334417, error = -0.16503% <- minimum K-value
a/b = 1/16, K(1/16)_exact = 0.8359937356, K(1/16)_approx = 0.8357161315, error = 0.033206%
Note the largest deviation of K(x) from 1 occurs when x = a/b = 0.118377, in which case K_min = 0.8273988891, and which is about 17.3% below unity. But even there the rational function approximation is only off of the correct minimum by 0.165%.
I made a graph comparing the values of K(a/b) coming from the numerical solution of the problem with the values coming from the rational function approximation to K(a/b). It is available at https://imgur.com/F3alwec . On that graphic image the black curve is for the numerical solution for K(a/b) vs a/b and the red curve is for the rational function approximation to K(a/b) vs a/b. Note in regions where the 2 curves coincide to a fraction of a pixel the black curve is on top and the red curve is hidden under it, and thus is not visible.
The values and form of the dimensionless fudge factor function K(a/b) is completely inaccessible by scaling arguments and by dimensional analysis even though the I ~ V^(3/2) form of the I vs V function and dimensional unit bearing constant factors *are* derivable results from such arguments.
The numerical solution for K(x) is found by numerically integrating the DE (with a 4th order Runge-Kutta algorithm with a tiny step size) for an auxiliary function w(σ) where σ = 1/x - 1 and K(x) = (σ^2)/((σ + 1)*w(σ)) and the w(σ) function is the unique solution to the DE
with initial condition w(0) = 0, w'(0) = 0, where w'(σ) = dw(σ)/dσ and w''(σ) = d^2w(σ)/dσ^2. The physical meaning of the dimensionless independent variable, σ, here is σ = r/a - 1 where r is some radial position between the electrodes so σ = 0 is located on the surface of the cathode (i.e. r = a) and σ = b/a - 1 is located on the surface of the anode (i.e. r = b). So the domain of the independent variable σ is 0 ≤ σ ≤ b/a - 1. The value of the dimensionless function w(σ) is proportional to the 3/2 power of the electric potential. The reason for considering a dimensionless function proportional to the 3/2 power of the electric potential is that it removes a power law singularity in the DE at the anode and so regularizes that point for future power series expansion purposes. Whenever a > 0 *both* the potential and electric field vanish at the cathode, and the DE has an initial condition w(0) = 0 and w'(0) = 0 at the cathode. The DE for the function w(σ) is integrated from σ = 0 up to σ = b/a - 1, and the value of w(b/a - 1) is used in the evaluation of K(a/b). Note when a is exactly zero the electric field does *not* vanish but rather *diverges* at the cathode. This singular behavior as a → 0 means we can't easily make smooth power series expansions about the filamentary cathode a -> 0 limit. That is why we instead make our expansion about the flat plate a → b limit in powers of σ. The power series solution for w(σ) in powers of σ (which really means in powers of b/a - 1 when it is eventually substituted back into the expression for K(a/b)) is
We know that in the a → 0 limit K(0) = 1. This implies that w(σ)/σ → 1 in the σ → ∞ limit. The simplest rational expression whose power series in σ matches the shown terms in above power series for w(σ) *and* has the property that w(σ)/σ → 1 in the σ → ∞ limit is
To make our rational function approximation for the K(x) function we substitute the above rational expression in for w(σ) in the equation relating K(x) in terms of w(σ). The result of this substitution is
which was displayed earlier above. To get the approximate fudge factor in Child's Law for a given a/b ratio just substitute a/b in for x in K(x)_approx. As we saw earlier this approximation is quite accurate over a the whole range of a/b values, typically with an error of a small fraction of a percent. Understandably is does the worst near the singular limit a → 0 (x → 0) since the rational function approximation only knows to go to the correct limit there, but it doesn't know *how* to go to that singular limit in terms of the behaviors of the rates of approach to it. Nevertheless, as the graph at https://imgur.com/F3alwec shows, the approximation does remarkably well, anyway, and we have answered the question as to the behavior of the fudge factor function K(x). The fact that it goes to the same limit of unity for both x = 0 and x = 1 is a fluke, not a feature of some deep connection between the cylindrical geometry with a filamentary cathode and a flat parallel plate geometry. This makes sense in hindsight since these two solvable limiting cases have such a wildly different set of charge, current, field & potential near the cathode.