Peirastes

On Physical Analogies

Contents

1. I. Introduction
2. II. Static System Parameters and Units
3. III. Dynamics
4. IV. Potential Energy vs Kinetic Energy
5. V. Dynamic System Parameters and Units

Warning to reader: Work in progress. May be redundant and scattered.

I. Introduction

During my time as a lab instructor, I tried to reconcile the terminology of "Electromotive Force" used often in textbooks and in our lab manual. You may already be aware that the EMF is not even a force! EMF has units of Joules/Coulomb - the same as electric potential, with units of energy per unit of charge. Indeed, electric potential energy can be found by multiplying the potential by the charge and the result will have units of energy (Joules). However, the name "Electromotive Force" implies that this physical parameter would have units of force – Newtons. Rather, it represents a potential. While working on my thesis, I attempted to rationalize the units of electric potential and gravitational potential to realize a comprehensible analogy of the latter to the former. Although the result may be rather obvious in hindsight (which is 20/20), I believe it worthwhile to document my findings and how I came to my results thus far, for there is much more to discern.

In summary, I am attempting to deduce a viable analogy between gravitation and electromagnetism that considers a set of basic parameters including displacement, velocity, acceleration, sources, fields, forces, energies, and potentials. This analogy is based on the writings of Oliver Heaviside and his work on describing gravitational waves and how they should relate directly to electromagnetic waves. In fact, Heaviside proposes that there exists a component field of the static gravitational field, or gravitic field, that is analogous to the magnetic field. This secondary gravitational field is called the cogravitational field. For Heaviside's namesake, this field will be referred to as the heavitic field, the gravitational analog to the magnetic field. To describe gravitational effects more fully, the term gravitoheavitic will be incorporated in the lexicon so to reflect the interaction of both fields and to reflect Heaviside's contribution to the study of gravity and gravitational waves.

I.1 Sources, Fields, and Forces

The source units of gravitation and electromagnetism are mass - with units of kilograms [kg] - and charge - with units of Coulombs [C]. Gravitational mass generates a radially inward gravitational field, or gravitic field, and the motion of mass generates a rotationally positive heavitic field. Electric charge generates a radially outward electric field, and the motion of a charge generates a rotationally positive magnetic field.

Gravitational masses interact with gravitoheavitic fields and electrical charges interact with electromagnetic fields. In electromagnetism, there are the electric and magnetic fields. In gravitation, there are the gravitic and heavitic fields. These fields are acceleration fields, with the gravitic or gravitational field, g, measured in meters per second squared [m/s^2] or Newtons per kilogram [N/kg] and the electric field, E, measured in Volts per meter [V/m] or Newtons per Coulomb [N/C]. Notice that the field dimensions are similarly related by force units per source units.

Gravitational masses in a gravitational field experience a force given by:

\(F = \text{mg}\),

where \(F\) is the force, \(m\) is the mass, and \(g\) is the acceleration due to the gravitational field. More generally, a mass experiencing any acceleration experiences a proportional force, expressed as:

\(F = \text{ma}\),

where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration.

Electrical charges in an electrical field experience a force given by:

\(F = \text{qE}\),

where \(F\) is the force, \(q\) is the charge, and \(E\) is the acceleration due to the electric field.

Notice that the force is similarly determined by multiplying the source with the field. Force, in both cases, is measured in Newtons [N]. In general, Sources interact with Fields and experience Forces.

I.4 Energies

Moving a source through a field with force requires energy. A relatable form of energy is work, which is simply the ability of a force to move things about. The work done by a force to move a source over some distance is given by:

\(W = Fd\cos\theta\),

where \(W\) is measured in Joules [J], \(F\) is the force measured in Newtons [N], \(d\) is the distance in meters [m], and \(\theta\) is the angle between the applied force and the line of action (direction of motion).

Similarly, the concept of potential energy is a form of energy and potential energy is also calculated using force and distance. Gravitational potential energy can be determined using the equation:

\(U = mgh = Fh\),

where \(U\) is the potential energy [J], \(m\) is the mass [kg], \(g\) is the gravitational acceleration due to the field [N/kg], \(h\) is the height of the mass from the zero-potential [m], and \(F\) is the force on the mass [N]. Dimensionally, the result of force multiplied with distance is Newton-meters [N-m], which is equivalent to Joules [J].

Electrical potential energy can be determined by:

\(U = qEs = Fs\),

where \(U\) is the potential energy [J], \(q\) is the charge [C], \(E\) is the electrical acceleration due to the field [N/C], \(s\) is the distance of the charge from the zero-potential [m], and \(F\) is the force on the charge [N]. Again, the dimensions of the force by the distance are units of energy (Joules).

I.5 Potentials

Sometimes the electrical potential energy is also described as,

\(U = Vq\),

where the electrical potential (voltage), \(V\), is defined as,

\(V = Es\).

This "charge-independent" value is known as electrical potential, or voltage, which has units of volts [V] or Joules per Coulomb [J/C]. The same can be done to gravitational potential energy to determine a "mass-independent" gravitational potential.

\(U = mgh\),

becomes,

\(U = mV\).

So, the gravitational potential is,

\(V = \frac{U}{m}\),

which becomes,

\(V = gh\).

In general, the potential is the potential energy per source unit [J/kg or J/C] and represents energy density, thus does not depend on mass or charge, respectively.

II. Static System Parameters and Units

Thus, we have the following table of field parameters in gravitation and electromagnetism:

Generalized Units	Field Parameter	Gravitation	Electromagnetism
[Source]	Source	m — Gravitational Mass [kg]	q — Electrical Charge [Coulomb]
[Field] = [Force/Source]	Field (Acceleration)	g — Gravitic Field [N/kg] [m/s2]	E — Electric Field [N/C] [V/m]
[Force] = [Source*Field]	Force	\(F = ma\) — Gravitic Force [N]	\(F = qE\) — Electric Force [N]
[Energy] = [Force*Displacement]	Potential Energy	\(U = mgh = Fh = Vm\) — Gravitational PE [J]	\(U = qEs = Fs = Vq\) — Electrical PE [J]
[Potential] = [Energy/Source]	Potential	\(V = U/m = gh\) — Gravitational Potential [J/kg]	\(V = U/q = Es\) — Electrical Potential [J/C] [V]

Drawing comparisons between the field parameters, some generalizations can be made. Most broadly, and most fundamental, is that Sources produce Fields and exert Forces. Forces are the products of Sources and Fields. The (potential) energy required to exert such forces is dependent on the displacement of the source in the field and is the product of that displacement with the exerted force. Potential is a measure of energy density and is the proportion of energy units to source units.

Dimensionally, Fields are the proportion of Force units to Source units. Energies are measured in Joules and Potentials are energy ratios dependent on the source (mass, charge). While the first four parameters can be rather intuitive (especially after comparing gravitation and electromagnetism), the fifth parameter - potential - is not as straightforward.

So then if voltages are now perhaps more "familiar", how is a gravitational potential relatable? What would be the gravitational equivalent/analog of voltage?

Recall, g [N/kg] and E [N/C] which are both written such that the units are given by the ratio of the force units to source units.

Thus, a source interacts with a field and experiences a force. The force is proportional to the magnitude of acceleration. For an arbitrary source, p, and acceleration field, A, the force experienced by the source in the field is given by

\[F = pA\]

To be more accurate, the field is produced by a parent source with which a child source interacts and experiences the force. Recall these equations of Newton and Coulomb:

\[F = - G\frac{\text{mM}}{R^{2}}\]

And

\[F = k\frac{\text{qQ}}{R^{2}}\]

Maybe more interesting is the relationship that can be drawn between units of potential. Until the units of the gravitational field were rewritten as force units per source units, they were simply [m/s^2] and identified as acceleration. When comparing units of the alternative expression for the electric field [V/m], the relationship is easily identified.

If gravitational acceleration units are rewritten as [(m²/s²)/m] and compared to [V/m], then Volts can be related to the units of gravitational potential as m²/s². To answer a previous question, gravitational and electrical potential are both expressions of energy units per source units, and thus, gravitational potential [m²/s²] can be thought of as gravitational voltage (though probably shouldn't be). Perhaps it would be more appropriate to refer it to as gravitational "pressure", as voltage is sometimes likened to electrical "pressure".

But what does a m²/s² represent physically? An acceleration over a distance, which is caused by a force and driven by energy. If we consider a voltage across two parallel plates, with one being grounded, there exist a series of parallel equipotentials. For example, a 1 V potential over a 1 mm distance, there will exist a potential of 0.5 V half way between the plates. At a distance of 0.25 mm from the ground plate, the voltage will be 0.25 V. Hence, the acceleration depends on the voltage gradient, or perhaps a gradient of active energy conversion – as the more energy/time available to accelerate the particle, the more energy is converted from potential energy to kinetic energy.

III. Dynamics

So far, only the static displacement of sources has been discussed. What about the movement of these sources at some velocity, v? Then,

Mass x Velocity = Momentum = [kg x m/s]

&

Charge x Velocity = [Coulomb x m/s]

So, what does this unit represent for electrodynamics?

Consider a resistive force, R, that is proportional to velocity by a constant such that

\[R = Bv\]

Let's look at the units of this constant, B,

\[N = \left[B\right]\frac{m}{s}\]

\[\left[B\right] = \frac{N \cdot s}{m} = \frac{kg \cdot m \cdot s}{s^{2} \cdot m} = \frac{\text{kg}}{s}\]

Then, B is a mass flow rate.

Let's now consider how this force relates to potential, where a is the net acceleration produced by the force, and h is the distance over which the force is applied.

\[V = ah = \frac{F}{m}h\]

If we substitute the force with the resistive force, R,

\[V = \frac{R}{m}h = \frac{Bv}{m}h = \frac{B}{m}\text{hv}\]

Written in the order of the last expression above, the potential is represented as the product of the ratio of mass flow rate to mass, distance, and velocity.

We should check dimensionally that this is indeed a potential,

\[\frac{m^{2}}{s^{2}} = \left(\frac{\frac{\text{kg}}{s}}{\text{kg}}\right)\left(m\right)\left(\frac{m}{s}\right)\]

And we see that this does in fact agree with the dimension of potential!

Let's rearrange the expression for potential to isolate the mass flow rate,

\[V = B\left(\frac{\text{hv}}{m}\right)\]

Doing this, the term in parentheses effectively represents the "resistance" of mass flow!

\[V = BR\]

This equation is effectively a gravitational "Ohm's Law", where B is the mass flow rate (mass-current), and R is the flow resistance (of which the units are strange). The units of the flow resistance will be referred to as "Gohs", for fun.

\[\left[\text{Gohs}\right] = \left[R\right] = \frac{m^{2}}{\text{kg} \cdot s}\]

With a possible gravitational analogy to Ohm's law (for ohmic devices), let's turn our attention to the three passive components of electric circuits and attempt to "derive" gravitational analogies to the resistor, capacitor, and inductor.

Since we have somewhat described the resistive component, with the flow resistance R, let's start with the capacitor. Consider how a spring is like a capacitor. Both are capable of "storing" charge given that they are not overworked. Both stored energies would like to discharge and induce a force proportional to the displacement of the energy sources (mass/charge). In short, both springs and capacitors store energy as a function of displacement of the source.

Comparing the two general force equations to highlight their similarities,

\(F = ma\)     \(F = qE\)

Starting with the relation between capacitance, charge, and voltage,

\(Q = CV\)     \(C = \frac{Q}{V}\)

By analogy, if H represents the mass-capacitance, we seek something of the form,

\(M = HV\)     \(H = \frac{M}{V}\)

Where H has the following units:

\[\frac{\text{kg} \cdot s^{2}}{m^{2}}\]

Let's check that this quantity holds in other familiar expressions. The Coulomb force could be expressed in terms of capacitance as,

\[F = \frac{\text{qQ}}{\text{dC}}\]

Which suggests there exists an analogous Newtonian force,

\[F = \frac{\text{mM}}{\text{dH}}\]

Checking dimensionally,

\[N = \frac{\text{kg}^{2}}{m\left(\frac{kg \cdot s^{2}}{m^{2}}\right)} = \frac{kg \cdot m}{s^{2}}\]

This checks out! The potential energy of a mass-like capacitor is,

\[PE = \frac{1}{2}MV = \frac{1}{2}HV^{2} = \frac{1}{2}H\left(\text{ad}\right)^{2}\]

Let's think about how this mass-capacitance, H, relates to the stiffness, k, of a spring.

\[PE = \frac{1}{2}kx^{2} = \frac{1}{2}HV^{2}\]

Solving for the stiffness,

\[k = \frac{HV^{2}}{x^{2}}\]

Then, the mass-capacitance can be expressed in the following forms:

\[H = \frac{m}{V} = \frac{kx^{2}}{V^{2}} = \frac{\text{mM}}{Fx}\]

Now we investigate mass inductance. An analogous description of potential using mass induction, \(N\):

\[V_{N} = N\ddot{m}\]

The units of mass induction are:

\[\left[N\right] = \frac{m^{2}}{\text{kg}}\]

Or, equally, Area/mass.

When switching between paradigms of mass and charge, let \(kg = C\). Then:

\[\left[L\right] = \frac{\text{kg} \cdot m^{2}}{C^{2}} = \frac{\text{kg} \cdot m^{2}}{\text{kg}^{2}} = \frac{m^{2}}{\text{kg}} = \left[N\right]\]

The dimension of stiffness can be described as:

\[\left[k\right] = \frac{N}{m} = \frac{J}{m^{2}} = \frac{\text{kg}}{s^{2}}\]

IV. Potential Energy vs Kinetic Energy

What I am struggling with comprehending/describing is HOW objects/sources obtain potential energy! And HOW that energy is converted to kinetic energy! Heaviside said that potential energy is just energy that is not known to be kinetic!

I've attempted to describe potential energy as energy obtained by a source displaced in a displacement field (gravitic/electric) and is determined by a displacement (space), while kinetic energy is energy obtained (or lost) through movement in a displacement field and is determined by velocity (time).

IV.1 Gravitational Energy (Gravitoheavitic)

Consider a ball on a desk. If we draw out coordinate axes such that the top of the desk is the "zero-potential", we can still apply the laws of motion as appropriately as if we had chosen the ground to be the location of "zero-potential". Now consider the ball raised to some height. The ball now "obtains" potential energy and, when released from rest, the energy will be converted into kinetic energy (with some - often negligible - losses).

This amount of potential energy is determined by the height the mass was raised with respect to the zero-potential, \(U = mgh\), where g is the acceleration produced by the displacement field. We also know that work is done, \(W = Fd\), so here, the energy needed to perform that work \(U = mgh\) can be recognized as \(U = Fh\) where F is the gravitational force on a mass, m, and h is the displacement height.

As the ball in the example is released, potential energy is converted into kinetic energy. As the acceleration field acts on the ball, which experiences a force (\(F = mg\)), it begins to fall and move with an increasing velocity over time. With this motion through the acceleration field, a cogravitational field is produced normal to the gravitational field and the direction of motion of the source.

IV.2 Electromagnetic Energy

Consider a charge displaced in an electric field. A positive source charge shown experiences a force, F, due to the electric field. As described previously, a source displaced in an acceleration field experiences a force. We know that \(F = qE\) and this is strikingly similar to \(F = ma\) as the force being the product of the source (mass/charge) and the field's acceleration.

IV.3 Generalized Energy

The acceleration field produced by a parent source has units that can also be written as the potential per distance, or in other words, the energy per source per distance.

I propose that:

Potential energy depends on position in an acceleration field.

Kinetic energy depends on motion in an acceleration field.

Still, this does nothing to exhaust Heaviside's statement that potential energy is energy that is not known to be kinetic.

To describe the motion of a source through a field, let's first consider how a source is set in motion. A change in state requires a force (acceleration). A source can only be accelerated by a force and a source only experiences a force when accelerated. To begin motion requires a (net) force!

Then, maybe, it would be appropriate to say that kinetic energy is the energy of a source experiencing a net force while potential energy is the energy of a source experiencing no net forces (inert)?

V. Dynamic System Parameters and Units

Generalized Units	Field Parameter	Gravitation	Electromagnetism
[Displacement]	Displacement	\(\Delta x = x - x_0 = d = h\) [meters]	\(\Delta s = s - s_0 = s\)
[Motion] = [Displacement/Time]	Velocity	\(v = d/t\) [m/s]	\(v = s/t\) [m/s]
[Source]	Source	m — Gravitational Mass [kg]	q — Electrical Charge [C]
[Acceleration] = [Field]	Static Field	g — Gravitic Field [N/kg]	E — Electric Field [N/C] [V/m]
[Force] = [Source*Field]	Static Force	\(F = mg\) — Gravitic Force [N]	\(F = qE\) — Electric Force [N]
[Energy] = [Force*Displacement]	Potential Energy	\(U = mgh = Fh = Vm\) [J]	\(U = qEs = Fs = Vq\) [J]
[Potential] = [Energy/Source]	Potential	\(V = U/m = gh\) [J/kg]	\(V = U/q = Es\) [J/C] [V]
[Field]	Dynamic Field	Heavitic Field	Magnetic Field
[Force]	Dynamic Force	Heavitic Force	Magnetic Force
[Energy] = [Source*Potential]	Kinetic Energy	\(K = \frac{1}{2}mv^2\) [J]	Electrical Kinetic Energy
[Power] = [Energy/Time]	Power	\(P = W/t\)	\(P = IE = RI^2\)
[Momentum] = [Source*Motion]	Momentum	\(p = mv = Ft\) [kg·m/s]	\(J = \rho v\) — Current Density [C/m2·s]
Capacitance [Source/Potential]	Capacitance (PE)	Spring: \(H = m/V\) [kg·s2/m2]	Capacitor: \(C = Q/V\) [s/Ohm]
Inductance [Area/Source]	Inductance (KE)	Flywheel (Inertia): \(I = A/m\)	Inductor [Henries]
	Conductance (Dissipation)	Dashpot/Damper (Friction)	Conductor [Siemens]

Unsorted Notes:

"Impedance ... is defined as the ratio of pressure to volume flow rate." Impedance is the ratio of pressure to flux. Pressure induces a flow rate proportional to impedance. Capacitance induces inductance proportional to conductance. A capacity of stored potential energy can induce kinetic energy proportional to the ability to conduce energy dissipation.