**PROOF THAT THE QUALITY FACTOR OF RESONANCE "Q" SCALES LIKE √L AND THEREFORE THAT THE FORCE/POWER ALSO SCALES LIKE √L****1) Force per power input of EMDrive and its relationship to a photon rocket**We start with the definition of Power as the time derivative of work, and therefore equal to the vector dot product of force times velocity:

For an ideal photon rocket with a perfectly collimated photon beam, the exhaust velocity (not the spaceship velocity !!!) is the speed of light c and therefore:

F*c = P

_{in}where P

_{in} is the Power input into the exhaust (

**"Power Input" here only stands for the power at this late stage, notice that there may be further losses from the power plant, coupling factor, etc.**). Therefore, for an ideal photon rocket, the force per input power is

(F / P

_{in})

_{photonRocket} = 1/c

Side note: for rockets exhausting particles-with-mass at speeds much lower than the speed of light, for example ion thrusters, this ratio is 2/v instead of 1/c, where v is the speed of the particle-having-mass (as propellant particles-with-mass, unlike photons, need to be accelerated to the exhaust speed, see

https://en.wikipedia.org/wiki/Spacecraft_propulsion#Power_to_thrust_ratio and

https://en.wikipedia.org/wiki/Specific_impulse#Energy_efficiency for the reason for the factor of 2, as E=(1/2)mv

^{2} instead of E=mc

^{2} https://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies ). Therefore, the efficiency (F / P

_{in}) for ion thrusters is much larger than the one for photon rockets (since v<< c, and hence 2/v>>1/c) and that's why this type of photon rocket has not seen, and is not envisioned to have, practical use.

Interestingly the force per input power for the EM Drive, according to all three different theories (McCulloch, Shawyer and Notsosureofit) can be expressed similarly as:

(F / P

_{in})

_{EMDrive} = (1/c) Q g

where Q is the quality factor of resonance and g is a dimensionless factor due to geometry, relative magnetic permeability, relative electric permittivity and mode shape, depending on the theory. So, the force per input power for an EM Drive is predicted to be superior to a photon rocket as follows:

(F / P

_{in})

_{EMDrive}/ (F / P

_{in})

_{photonRocket} = Q g

**in other words, the theoretical outperformance of the EMDrive is speculated to be due to just the quality of resonance Q and the dimensionless factor g.**For the purpose of this discussion we will avoid dealing with the strange consequences of these theories regarding conservation of momentum and conservation of energy issues inherent to the concept of proposing a closed resonant electromagnetic cavity for space propulsion.

**2) Geometric factor "g" for different theories****2a) McCulloch**McCulloch has presented a number of simple formulas for the EMDrive (

http://www.ptep-online.com/index_files/2015/PP-40-15.PDF), all having the general form

(F / P

_{in})

_{EMDrive} = (1/c) Q g

The simplest of which has the following simple definition for the dimensionless factor "g":

g=(L/D

_{s} - L/D

_{b})

where:

L = length of fustrum of a cone, measured perpendicular to the end faces

D

_{s}=diameter of small end of the fustrum of a cone

D

_{b}=diameter of big end of the fustrum of a cone

So, it is evident that for this formula from McCulloch, the factor "g" is a dimensionless factor that only depends on the geometrical ratios L/D

_{s} and L/D

_{b}:

g

_{McCulloch} = g (L/D

_{s},L/D

_{b})

It is also obvious that if one scales the EM Drive geometry such that the geometrical ratios L/D

_{s} and L/D

_{b} are kept constant, that the dimensionless factor "g" will remain constant in McCulloch's equation.

**2b) Shawyer**Shawyer has presented a formula (

http://www.emdrive.com/theorypaper9-4.pdf ) for the EM Drive where the dimensionless factor "g" is defined as follows:

g = 2 D

_{f}where D

_{f} is a dimensionless factor called the "Design Factor" by Shawyer, and where D

_{f} is a function of the diameters and in addition it is also a function of the relative magnetic permeability and the relative electric permittivity, as well as the natural frequency of resonance:

g

_{Shawyer} = g(D

_{b}/D

_{s},L/D

_{b},μ

_{r},ε

_{r},m,n,p)

where the diameters of the fustrum of a cone appear explicitly in his formula for the "design factor" and where the length and the mode shape quantum numbers appear only implicitly because the design factor is dependent on the natural frequency at which resonance with a particular mode shape occurs.

It is simple to show that if one scales the EM Drive geometry such that the geometrical ratios L/D

_{s} and L/D

_{b}, and the material properties μ

_{r},ε

_{r} are kept constant, and the mode shape is kept the same, that the dimensionless factor "g" will remain constant in Shawyer's equation.

**2c) Notsosureofit**Notsosureofit has presented a formula for the EM Drive (

http://emdrive.wiki/@notsosureofit_Hypothesis) where the dimensionless factor "g" is defined as follows:

g=(Ψ

_{mn}^{2}/(4π

^{3}))(c/f

_{mnp})

^{3}(1/L)(1/(D

_{s})

^{2}-1/(D

_{b})

^{2})

where

Ψ

_{mn}= X

_{mn} (the zeros of the cylindrical Bessel functions) for TM modes

Ψ

_{mn}= X'

_{mn} (the zeros of the first derivative of the cylindrical Bessel functions) for TE modes

(Side note: This link is an excellent source for the numerical values of X

_{mn} and of X'

_{mn} for m<11 and n<6:

http://wwwal.kuicr.kyoto-u.ac.jp/www/accelerator/a4/besselroot.htmlx )

Therefore, it can be shown that the "g" factor in Notsosureofit's hypothesis is a function of the geometrical ratios

g

_{Notsosureofit} = g(L/D

_{s},L/D

_{b},μ

_{r},ε

_{r},m,n,p)

how this is so, will be shown in detail in the next section.

**3) Natural frequency scaling**For simplicity, since the truncated cone resonant cavities tested by NASA, Shawyer, Tajmar, and others have all been close to a cylindrical cavity, we will derive the scaling relationship for the natural frequencies of a cylindrical cavity, but this can also be done with the more complicated equations for a truncated cone (which instead of cylindrical Bessel functions are expressed in terms of spherical Bessel functions, and instead of harmonic (cosine) functions in terms of Associated Legendre functions). The reason why all EM Drive experiments have been performed up to now with EM Drive geometries close to a cylindrical cavity is because experimenters have tried to follow Shawyer's strange prescription that the small diameter of the truncated cone should be larger than the cut-off frequency for an open, constant-cross-section waveguide having the same diameter (although the EM Drive is a closed cavity, and not an open waveguide, and it is well-known that such cut-off equations are inapplicable to closed cavities). This prescription forbids geometries of truncated cones where the small diameter is much different from the large diameter. Therefore it turns out that one can use a mean radius

R = (D

_{s} + D

_{b})/4

to model the fustrum of a cone cavity as a cylindrical cavity, having natural frequencies

f

_{mnp}=(c/R) a

_{mnp}where c is the speed of light, R is the previously defined mean radius and where m,n,p are the so called "quantum numbers" defining the mode shape, where m is the integer related to the circumferential direction, n is the integer related to the polar radial direction and p is the integer related to the longitudinal axial direction.

And where

a

_{mnp}= √((( Ψ

_{mn}/ π)

^{2}+(p R/L)

^{2})/(4 μ

_{r} ε

_{r}))

It is also trivial to show that since

R = (D

_{s} + D

_{b})/4

then

R/L =(D

_{s}/L + D

_{b}/L)/4

hence

a

_{mnp}= a

_{mnp}( L/D

_{s},L/D

_{b},μ

_{r},ε

_{r},m,n,p)

and that for constant geometrical ratios, constant medium properties μ

_{r},ε

_{r}, and for the same mode shape m,n,p, a

_{mnp} will remain constant and hence that

**the frequency will scale like the inverse of any geometrical dimension**:

f

_{mnp}=(c/R) constant

=(c/L) constant

=(c/ D

_{b}) constant

=(c/ D

_{s}) constant

**3a) Proof that Notsosureofit's dimensionless factor is constant for constant geometrical ratio, constant medium properties and constant mode shape**Returning to Notsosureofit's dimensionless factor expression in point 2c:

g=(Ψ

_{mn}^{2}/(4π

^{3}))(c/f

_{mnp})

^{3}(1/L)(1/(D

_{s})

^{2}-1/(D

_{b})

^{2})

and replacing the frequency expression:

f

_{mnp}=(c/R) a

_{mnp}one obtains:

g=(Ψ

_{mn}^{2}/(4π

^{3}))(R/a

_{mnp})

^{3}(1/L)(1/(D

_{s})

^{2}-1/(D

_{b})

^{2})

and therefore,

g

_{Notsosureofit} = g(L/D

_{s},L/D

_{b},μ

_{r},ε

_{r},m,n,p)

since:

R/L =(D

_{s}/L + D

_{b}/L)/4

(R/D

_{s})

^{2}=((1/4)(1+D

_{b}/D

_{s}))

^{2}(R/D

_{b})

^{2}=((1/4)(1+D

_{s}/D

_{b}))

^{2}(D

_{b}/D

_{s})= (L/D

_{s})/(L/D

_{b})

Therefore for constant geometrical ratios, constant medium properties μ

_{r},ε

_{r}, and for the same mode shape m,n,p,

**the dimensionless factor g will remain constant**. It is trivial to show the same result for Shawyer's design factor, and hence for the dimensionless factor g in Shawyer's expression.

**So, in general we can state that all theoretical expressions, McCulloch's, Shawyer's and Notsosureofit, are such that the dimensionless factor g will remain constant for constant geometrical ratios, constant medium properties μ**_{r},ε_{r}, and for the same mode shape m,n,p.

**4) Quality of resonance (Q) scaling**The definition of quality of resonance factor (Q) can be stated as follows (

https://en.wikipedia.org/wiki/Q_factor#Definition_of_the_quality_factor):

Q ≝ ω EnergyStored /PowerLoss

where

ω = angular frequency

EnergyStored =∫Electromagnetic Energy Density dV

PowerLoss = ((ω δ) /2) (∫ Electromagnetic Energy Density dA)

= R

_{s} (∫ Electromagnetic Energy Density dA)/ μ

= ρ (∫ Electromagnetic Energy Density dA)/ (μ δ)

where

R

_{s} = "surface resistance"

= ρ / δ

ρ = resistivity of the interior wall of the EM Drive resonant cavity

μ = magnetic permeability of the interior wall of the EM Drive resonant cavity

= μ

_{o}μ

_{r}δ =skin depth (the penetration depth of the electromagnetic energy into the interior metal wall)

(

https://en.wikipedia.org/wiki/Skin_effect)

in general, for arbitrary frequencies, the skin depth is:

where

ε = electric permittivity of the interior wall of the EM Drive resonant cavity

= ε

_{o}ε

_{r}At angular frequencies ω much below 1/(ρε), for example, in the case of copper, for frequencies much below exahertz (10^9 GHz, the range of hard X-rays and Gamma rays), the skin depth can be expressed as follows:

Now, using the fact that

PowerLoss =((ω δ) /2) (∫ ElectromagneticEnergy dA)

one immediately obtains:

Q=(2/SkinDepth)( ∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

Alternatively one can arrive at the same result, using the formula for power loss that depends on the "surface resistance" R

_{s}:

PowerLoss = R

_{s} (∫ Electromagnetic Energy Density dA)/ μ

PowerLoss = ρ (∫ Electromagnetic Energy Density dA)/ (μ δ)

one gets:

Q = ω μ (∫Electromagnetic Energy Density dV)/ (R

_{s} ∫ Electromagnetic Energy Density dA)

Q = ω μ δ (∫Electromagnetic Energy Density dV)/ (ρ ∫ Electromagnetic Energy Density dA)

and using the fact (at angular frequencies ω much below 1/(ρε) ) that the angular frequency ω is a function of the square of the skin depth δ:

ω = 2 ρ / (μ δ

^{2})

it is straightforward to show that the quality of resonance Q is:

Q=(2/SkinDepth)( ∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

the electromagnetic energy density integrated over the cavity volume, divided by the electromagnetic energy density integrated over the cavity surface area, divided by the skin depth.

**4a) Skin depth scaling**At frequencies much below 1/(ρε) the skin depth can be expressed as

SkinDepth = √(ρ/(μ π f

_{mnp}))

where

ρ = resistivity of the interior wall of the EM Drive resonant cavity

ε = electric permittivity of the interior wall of the EM Drive resonant cavity

= ε

_{o}ε

_{r}μ = magnetic permeability of the interior wall of the EM Drive resonant cavity

= μ

_{o}μ

_{r}f

_{mnp} = resonant frequency at mode shape m,n,p

= ω

_{mnp}/(2π)

Plugging in the expression for the frequency

f

_{mnp}=(c/R) a

_{mnp}into the skin depth expression, results in the following expression:

SkinDepth = √R√(ρ/(μ π c a

_{mnp}))

or, using the previously derived expressions for a

_{mnp} one concludes that

**the skin depth scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ**_{r},ε_{r}, and for the same mode shape m,n,p.

In other words, for increasing dimensions of the cavity, preserving all geometrical ratios, and keeping material properties constant and for the same mode shape, the skin depth will increase with the square root of the dimension, while the frequency will decrease, as the inverse of the dimension.

**4b) Quality of resonance (Q) scaling**Having revealed the scaling law for the skin depth, what now remains to be shown is the scaling for the energy integral ratio in the expression for Q:

Q=(2/SkinDepth)(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

The expressions under the integrals are dependent on each mode shape, as the electromagnetic energy distribution depends on mode shape, of course. However, we can notice that the lowest mode shapes (those with low values of m,n,p, for example TE012, TM212) have been of interest in the EM Drive experiments so far. So, for simplification purposes we can assume that the distribution of the electromagnetic field is of low order, and hence not that much variable throughout the cavity, for low mnp number mode shapes (for example m=0 means a constant distribution in the azimuthal circumferential direction of the cavity). Under this assumption one can (for approximation purposes) take the energy out of the integral:

(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA) ~

~ (Electromagnetic Energy Density /Electromagnetic Energy Density) (∫dV/ ∫ dA )

~ InteriorVolume/InteriorSurfaceArea

~ π R

^{2}L/(2 π R (R+L) )

~ R/(2(1+R/L))

and substituting this and the previously found scaling law for the skin depth, into the expression for the quality of resonance factor Q, leads to:

Q=(2/SkinDepth)(∫Electromagnetic Energy Density dV/ ∫ Electromagnetic Energy Density dA)

~(2/(√R√(ρ/(μ π c a

_{mnp})))) R/(2(1+R/L))

~ √R b

where the factor b is:

b = (1/((1+R/L)√(ρ/(μ π c a

_{mnp}))))

or, using the previously derived expressions for a

_{mnp} one concludes that

**the quality of resonance (Q) scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ**_{r},ε_{r}, and for the same mode shape m,n,p.

In other words, for increasing dimensions of the cavity, preserving all geometrical ratios, and keeping material properties constant and for the same mode shape, the quality of resonance (Q) will increase with the square root of the dimension, also the skin depth will increase with the square root of the dimension, while the frequency will decrease, as the inverse of the dimension.

Furthermore, we previously proved that all three theories for the EM Drive (McCulloch, Shawyer and Notsosureofit) have expressions for the force/inputPower to be proportional to the quality of factor Q times a dimensionless factor g:

(F / P

_{in})

_{EMDrive}/ (F / P

_{in})

_{photonRocket} = Q g

(F / P

_{in})

_{EMDrive} = (1/c) Q g

and we previously proved that the dimensionless factor g (for all three theories: McCulloch, Shawyer and Notsosureofit) remains perfectly constant for constant geometrical ratios, constant medium properties μ

_{r},ε

_{r}, and for the same mode shape m,n,p.

**Therefore one concludes that the force per input Power (for all three theories: McCulloch, Shawyer and Notsosureofit) scales like the square root of any geometrical dimension, for constant resistivity and magnetic permeability of the interior wall of the cavity and for constant geometrical ratios, constant medium properties μ**_{r},ε_{r}, and for the same mode shape m,n,p.

In other words, to maximize the force per input power, according to all three theories: (McCulloch, Shawyer and Notsosureofit) the most efficient EM Drive would be as large as possible, this being due to the fact that the quality of factor of resonance Q (all else being equal) scales like the square root of the geometrical dimensions.

Small cavity EM Drive's (all else being equal) are predicted to have smaller quality of resonance Q and therefore smaller force/inputPower.

It is not clear whether this has been known to EM Drive experimenters, given the fact that the recent experiments by Prof. Tajmar at TU Dresden, Germany, (under advise from Roger Shawyer according to the report) were performed with a much smaller EM Drive, and the fact that there are several EM Drive researchers discussing really tiny EM Drives (as the group in Aachen, Germany) for use in CubeSats. Such EM Drives are predicted to be much more inefficient, having substantially lower force/inputPower.