Abstract

A photoelectric photometer designed for the measurement of absolute turbidity, dissymmetry, and depolarization of dilute solutions of high molecular weight materials, and hence determination of their molecular weights, is described. The photometer comprises essentially a monochromatic parallel primary beam of radiation, a six-sided scattering cell for measurements at 0°, 45°, 90°, and 135°, a multiplier phototube and galvanometer, a standard opal glass diffusor, and removable polarizer and analyzer. Turbidity is determined in terms of a ratio of deflections for the 90° scattering and for the primary beam reduced in intensity by neutral filters and diffused by an opal glass plate. Working relationships leading to determination of absolute turbidity are developed. These relationships include corrections for refraction and reflection effects, and for imperfect diffusion by the opal glass. The latter is evaluated by comparison of the opal glass with reflecting diffusors corrected for specular component of reflectance. The response of some multiplier photo-tubes is shown to be dependent on the plane of polarization of the incident radiation. Data illustrating the performance of the photometer include comparison of molecular weights of polystyrene fractions, beta-lactoglobulin, bovine serum albumin, lysozyme, sucrose octaacetate, Raleigh’s ratio and depolarization for benzene, turbidity of a “standard” polystyrene, and particle size of a GR-S latex, with data obtained by other methods or other investigators.

© 1950 Optical Society of America

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References

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  1. P. Debye, J. App. Phys. 15, 338 (1944).
    [Crossref]
  2. P. Debye, J. Phys. and Colloid Chem. 51, 18 (1947).
    [Crossref]
  3. See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
    [Crossref] [PubMed]
  4. F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The absolute determination of scattered intensity” (January15, 1945).
  5. F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The determination of molecular weights and particle sizes by light scattering” (March1, 1945).
  6. C. I. Carr, dissertation, University of California, “An investigation of the molecular weights of molecules in solution by the light scattering method” (September, 1949). See also P. Outer, C. I. Carr, and B. H. Zimm, J. Chem. Phys. 18, 830 (1950).
    [Crossref]
  7. R. S. Stein and P. Doty, J. Am. Chem. Soc. 68, 159 (1946).
    [Crossref]
  8. P. P. Debye, J. App. Phys. 17, 392 (1946).
    [Crossref]
  9. R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
    [Crossref]
  10. B. H. Zimm, J. Chem. Phys. 16, 1099 (1948).
    [Crossref]
  11. R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
    [Crossref]
  12. H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
    [Crossref]
  13. P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).
    [Crossref]
  14. R. Speiser and B. A. Brice, J. Opt. Soc. Am. 36, 364A (1946).
  15. B. T. Barnes and W. E. Forsythe, J. Opt. Soc. Am. 27, 83 (1937).
    [Crossref]
  16. Radiation terms and symbols are essentially those proposed by the Committee on Colorimetry of the Optical Society of America, J. Opt. Soc. Am. 34, 245 (1944).
  17. See also H.-J. Helwig, Optik 5, 419 (1949).
  18. A. H. Taylor, (July28, 1920).
  19. V. G. W. Harrison, Proc. Phys. Soc., London 58, 408 (1946).
    [Crossref]
  20. J. S. Preston, Trans. Opt. Soc., London 31, 15 (1929–30).
    [Crossref]
  21. I. G. Priest and J. O. Riley, J. Opt. Soc. Am. 20, 156–157A (1930).
  22. R. S. Hunter, Paper Trade J. 126, 47 (1948).
  23. J. W. Ryde, Proc. Roy. Soc., London 131A, 451 (1931); J. W. Ryde and B. S. Cooper, ibid.,464; J. W. Ryde and B. S. Cooper, J. Soc. Glass Tech. 16, 408, 430 (1932).
    [Crossref]
  24. M. Luckiesh, Elec. World 60, 1040 (1912); Elec World 61, 883 (1913).
  25. P. Peyrot, Comptes Rendus 203, 1512 (1936).
  26. T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
    [Crossref]
  27. B. H. Zimm and I. Myerson, J. Am. Chem. Soc. 68, 911 (1946).
    [Crossref]
  28. B. A. Brice and R. Speiser, J. Opt. Soc. Am. 36, 363A (1946). Complete details will be published.
  29. M. Halwer, J. Am. Chem. Soc. 70, 3985 (1948). The coefficient of c2 in the equation should be 0.02831.
    [Crossref]

1949 (3)

R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
[Crossref]

H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
[Crossref]

See also H.-J. Helwig, Optik 5, 419 (1949).

1948 (4)

R. S. Hunter, Paper Trade J. 126, 47 (1948).

R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
[Crossref]

B. H. Zimm, J. Chem. Phys. 16, 1099 (1948).
[Crossref]

M. Halwer, J. Am. Chem. Soc. 70, 3985 (1948). The coefficient of c2 in the equation should be 0.02831.
[Crossref]

1947 (1)

P. Debye, J. Phys. and Colloid Chem. 51, 18 (1947).
[Crossref]

1946 (6)

R. S. Stein and P. Doty, J. Am. Chem. Soc. 68, 159 (1946).
[Crossref]

P. P. Debye, J. App. Phys. 17, 392 (1946).
[Crossref]

V. G. W. Harrison, Proc. Phys. Soc., London 58, 408 (1946).
[Crossref]

R. Speiser and B. A. Brice, J. Opt. Soc. Am. 36, 364A (1946).

B. H. Zimm and I. Myerson, J. Am. Chem. Soc. 68, 911 (1946).
[Crossref]

B. A. Brice and R. Speiser, J. Opt. Soc. Am. 36, 363A (1946). Complete details will be published.

1945 (1)

See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
[Crossref] [PubMed]

1944 (2)

1943 (1)

T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
[Crossref]

1937 (1)

1936 (1)

P. Peyrot, Comptes Rendus 203, 1512 (1936).

1935 (1)

P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).
[Crossref]

1931 (1)

J. W. Ryde, Proc. Roy. Soc., London 131A, 451 (1931); J. W. Ryde and B. S. Cooper, ibid.,464; J. W. Ryde and B. S. Cooper, J. Soc. Glass Tech. 16, 408, 430 (1932).
[Crossref]

1930 (1)

I. G. Priest and J. O. Riley, J. Opt. Soc. Am. 20, 156–157A (1930).

1912 (1)

M. Luckiesh, Elec. World 60, 1040 (1912); Elec World 61, 883 (1913).

Alfrey, T.

T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
[Crossref]

Badger, R. M.

R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
[Crossref]

Barnes, B. T.

Bartovics, A.

T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
[Crossref]

Billmeyer, F. W.

F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The absolute determination of scattered intensity” (January15, 1945).

F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The determination of molecular weights and particle sizes by light scattering” (March1, 1945).

Blaker, R. H.

R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
[Crossref]

Brice, B. A.

R. Speiser and B. A. Brice, J. Opt. Soc. Am. 36, 364A (1946).

B. A. Brice and R. Speiser, J. Opt. Soc. Am. 36, 363A (1946). Complete details will be published.

Brosteaux, J.

P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).
[Crossref]

Carr, C. I.

C. I. Carr, dissertation, University of California, “An investigation of the molecular weights of molecules in solution by the light scattering method” (September, 1949). See also P. Outer, C. I. Carr, and B. H. Zimm, J. Chem. Phys. 18, 830 (1950).
[Crossref]

Debye, P.

P. Debye, J. Phys. and Colloid Chem. 51, 18 (1947).
[Crossref]

P. Debye, J. App. Phys. 15, 338 (1944).
[Crossref]

Debye, P. P.

P. P. Debye, J. App. Phys. 17, 392 (1946).
[Crossref]

Dixon, J. K.

R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
[Crossref]

Doty, P.

R. S. Stein and P. Doty, J. Am. Chem. Soc. 68, 159 (1946).
[Crossref]

See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
[Crossref] [PubMed]

Forsythe, W. E.

Gilman, T. S.

R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
[Crossref]

Hadow, H. J.

H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
[Crossref]

Halwer, M.

M. Halwer, J. Am. Chem. Soc. 70, 3985 (1948). The coefficient of c2 in the equation should be 0.02831.
[Crossref]

Harrison, V. G. W.

V. G. W. Harrison, Proc. Phys. Soc., London 58, 408 (1946).
[Crossref]

Helwig, H.-J.

See also H.-J. Helwig, Optik 5, 419 (1949).

Hunter, R. S.

R. S. Hunter, Paper Trade J. 126, 47 (1948).

Hyde, J. C.

H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
[Crossref]

Luckiesh, M.

M. Luckiesh, Elec. World 60, 1040 (1912); Elec World 61, 883 (1913).

Mariner, T.

R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
[Crossref]

Mark, H.

T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
[Crossref]

Myerson, I.

B. H. Zimm and I. Myerson, J. Am. Chem. Soc. 68, 911 (1946).
[Crossref]

Peyrot, P.

P. Peyrot, Comptes Rendus 203, 1512 (1936).

Preston, J. S.

J. S. Preston, Trans. Opt. Soc., London 31, 15 (1929–30).
[Crossref]

Priest, I. G.

I. G. Priest and J. O. Riley, J. Opt. Soc. Am. 20, 156–157A (1930).

Putzeys, P.

P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).
[Crossref]

Riley, J. O.

I. G. Priest and J. O. Riley, J. Opt. Soc. Am. 20, 156–157A (1930).

Ryde, J. W.

J. W. Ryde, Proc. Roy. Soc., London 131A, 451 (1931); J. W. Ryde and B. S. Cooper, ibid.,464; J. W. Ryde and B. S. Cooper, J. Soc. Glass Tech. 16, 408, 430 (1932).
[Crossref]

Sheffer, H.

H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
[Crossref]

Speiser, R.

R. Speiser and B. A. Brice, J. Opt. Soc. Am. 36, 364A (1946).

B. A. Brice and R. Speiser, J. Opt. Soc. Am. 36, 363A (1946). Complete details will be published.

Stamm, R. F.

R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
[Crossref]

Stein, R. S.

R. S. Stein and P. Doty, J. Am. Chem. Soc. 68, 159 (1946).
[Crossref]

See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
[Crossref] [PubMed]

Taylor, A. H.

A. H. Taylor, (July28, 1920).

Zimm, B. H.

B. H. Zimm, J. Chem. Phys. 16, 1099 (1948).
[Crossref]

B. H. Zimm and I. Myerson, J. Am. Chem. Soc. 68, 911 (1946).
[Crossref]

See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
[Crossref] [PubMed]

Can. J. Research (1)

H. J. Hadow, H. Sheffer, and J. C. Hyde, Can. J. Research 27B, 791 (1949).
[Crossref]

Comptes Rendus (1)

P. Peyrot, Comptes Rendus 203, 1512 (1936).

Elec. World (1)

M. Luckiesh, Elec. World 60, 1040 (1912); Elec World 61, 883 (1913).

J. Am. Chem. Soc. (4)

M. Halwer, J. Am. Chem. Soc. 70, 3985 (1948). The coefficient of c2 in the equation should be 0.02831.
[Crossref]

T. Alfrey, A. Bartovics, and H. Mark, J. Am. Chem. Soc. 65, 2319 (1943).
[Crossref]

B. H. Zimm and I. Myerson, J. Am. Chem. Soc. 68, 911 (1946).
[Crossref]

R. S. Stein and P. Doty, J. Am. Chem. Soc. 68, 159 (1946).
[Crossref]

J. App. Phys. (2)

P. P. Debye, J. App. Phys. 17, 392 (1946).
[Crossref]

P. Debye, J. App. Phys. 15, 338 (1944).
[Crossref]

J. Chem. Phys. (2)

R. F. Stamm, T. Mariner, and J. K. Dixon, J. Chem. Phys. 16, 423 (1948).
[Crossref]

B. H. Zimm, J. Chem. Phys. 16, 1099 (1948).
[Crossref]

J. Opt. Soc. Am. (5)

R. Speiser and B. A. Brice, J. Opt. Soc. Am. 36, 364A (1946).

B. T. Barnes and W. E. Forsythe, J. Opt. Soc. Am. 27, 83 (1937).
[Crossref]

Radiation terms and symbols are essentially those proposed by the Committee on Colorimetry of the Optical Society of America, J. Opt. Soc. Am. 34, 245 (1944).

B. A. Brice and R. Speiser, J. Opt. Soc. Am. 36, 363A (1946). Complete details will be published.

I. G. Priest and J. O. Riley, J. Opt. Soc. Am. 20, 156–157A (1930).

J. Phys. and Colloid Chem. (2)

R. H. Blaker, R. M. Badger, and T. S. Gilman, J. Phys. and Colloid Chem. 53, 794 (1949).
[Crossref]

P. Debye, J. Phys. and Colloid Chem. 51, 18 (1947).
[Crossref]

Optik (1)

See also H.-J. Helwig, Optik 5, 419 (1949).

Paper Trade J. (1)

R. S. Hunter, Paper Trade J. 126, 47 (1948).

Polymer Bull. (1)

See, for example, B. H. Zimm, R. S. Stein, and P. Doty, Polymer Bull. 1, 90 (1945); G. Oster, Chem. Rev. 43, 319 (1948); R. H. Ewart, C. P. Roe, P. Debye, and J. R. McCartney, J. Chem. Phys. 14, 686 (1946); P. M. Doty, B. H. Zimm, and H. Mark, J. Chem. Phys. 13, 159 (1946).
[Crossref] [PubMed]

Proc. Phys. Soc., London (1)

V. G. W. Harrison, Proc. Phys. Soc., London 58, 408 (1946).
[Crossref]

Proc. Roy. Soc., London (1)

J. W. Ryde, Proc. Roy. Soc., London 131A, 451 (1931); J. W. Ryde and B. S. Cooper, ibid.,464; J. W. Ryde and B. S. Cooper, J. Soc. Glass Tech. 16, 408, 430 (1932).
[Crossref]

Trans. Faraday Soc. (1)

P. Putzeys and J. Brosteaux, Trans. Faraday Soc. 31, 1314 (1935).
[Crossref]

Trans. Opt. Soc., London (1)

J. S. Preston, Trans. Opt. Soc., London 31, 15 (1929–30).
[Crossref]

Other (4)

A. H. Taylor, (July28, 1920).

F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The absolute determination of scattered intensity” (January15, 1945).

F. W. Billmeyer, Rubber Reserve Company, Technical Report, “The determination of molecular weights and particle sizes by light scattering” (March1, 1945).

C. I. Carr, dissertation, University of California, “An investigation of the molecular weights of molecules in solution by the light scattering method” (September, 1949). See also P. Outer, C. I. Carr, and B. H. Zimm, J. Chem. Phys. 18, 830 (1950).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Photometer. A shows lamp housing; light-proof Duralumin box, 24×16×11 in., containing the scattering system, receiver, and power supply; galvanometer, scattering cells; removable polarizer and analyzer; and opal glass standard diffusor; B (lids open and front panel removed) shows receiver set at 45° to primary beam, graduated disk, dissymmetry cell, and working standard.

Fig. 2
Fig. 2

Diagrammatic sketch of optical system: L, mercury lamp; F1, monochromatic filters; S, camera shutter; F2, neutral filters mounted on a sliding carriage; L1, achromatic lens (ƒ=122 mm); L2, cylindrical lens (ƒ=200 mm) with axis horizontal; SC, scattering cell (40×40 mm) on fixed table; D, graduated disk attached to rotatable arm, A, carrying removable working standard, W, and the receiving system (shown in 90° position, 0° position dotted); limiting rectangular diaphragms in primary beam, D1 (13 mm wide × 11 mm high), D2 (15×15 mm), and D3 (12×15 mm); limiting rectangular diaphragms in receiving system; D4 (3.1×6.4 mm) and D5 (7.4×22 mm); O, opal glass depolarizing diffusor close to the multiplier photo-tube, M; P1 and P2, positions of demountable polarizer and analyzer; T, blackened removable tube serving as a light trap and as a means of aligning parts; H, covered peephole; B1, scattering compartment; B2, power supply compartment.

Fig. 3
Fig. 3

Diagram of electrical circuit. R, electronic voltage regulator; AH-3, mercury lamp with autotransformer, T1; S, switch, d.p.s.t.; F, fuses, 2a, 250 v; T2, variable transformer; T3, transformer, 800 v, r.m.s.; T4, transformer, 2.5 v, 5a; L, pilot lamp; 2X2-A, rectifier tube; C1 and C2, condensers, 1 μf, 1500 v; R1R10, fixed resistors, 125,000 ohms (matched ±2 percent), 1 2 watt; R11, 5000 ohms, 2 watts; 1P21, multiplier. photo-tube; G, galvanometer; R12, galvanometer shunt, 15,000 ohms.

Fig. 4
Fig. 4

Geometrical relationships (horizontal section) for scattering solution viewed at 90° and standard diffusor viewed at 0°. SC, scattering cell; D4 and D5, receiver diaphragms; O, diffusing opal glass near photo-tube M; GS, standard opal glass diffusor; C, compensating cell. In the apparatus (dimensions in mm), b=22, h=12, L=15, W=10, r=89 (102 when opal glass, O, is removed). The angle ϕ/2=5.7°.

Fig. 5
Fig. 5

Refraction effects: foreshortening of radius r to rs, and spreading of solid angle from πθ2 to πθs2.

Fig. 6
Fig. 6

Plots of Hc/τ versus concentration, for wave-length 436 mμ, for three fractions of polystyrene in methyl ethyl ketone.

Fig. 7
Fig. 7

Plots of Hc/τ versus concentration for wave-lengths 436 and 546 mμ for sucrose octaacetate.

Tables (6)

Tables Icon

Table I Calibration data for turbidity equation.

Tables Icon

Table II Comparison of reflecting diffusors (i=−45°, α=45°) with solid opal glass transmitting diffusor (546 mμ).

Tables Icon

Table III Transmittances of neutral filters (F, photometer; T, spectrophotometer).

Tables Icon

Table IV Dependence of multiplier photo-tube response on plane of polarization. Effectiveness of opal glass as depolarizer.

Tables Icon

Table V Dissymmetry coefficients, q, depolarization, ρu, turbidity correction factors, and refractive index increments.

Tables Icon

Table VI Comparison of light-scattering results with data by other methods or other investigators.

Equations (26)

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H c / τ = ( 1 / M ) + ( 2 B / R T ) c +
H = 32 π 3 n 0 2 ( n - n 0 ) 2 / c 2 3 λ 4 N 0 ,
q = ( G 45 / G 135 ) - 1 ,
ρ u = F H u / V u .
ρ v = F H v / V v .
W s W = 1 - b r - x n - 1 n .
r s = r ( 1 - b r n - 1 n ) .
I I s = π r s 2 θ s 2 π r 2 θ 2 = ( 1 - b r n - 1 n ) 2 n 2 .
P = 0 π J 0 r 2 1 + cos 2 θ 2 2 π r ( sin θ ) r d θ = 8 3 π J 0 = 16 3 π J 90 ,
P = P 0 x τ = I 0 V τ
J 90 = ( 3 / 16 π ) I 0 τ V
I = J 90 / r 2 = ( 3 / 16 π r 2 ) I 0 τ V .
I s = 3 ( 1.045 ) I 0 τ ( W s L h ) T s 16 π r 2 [ 1 - ( b / r ) ( n - 1 ) / n ] 2 n 2 .
P = 0 π / 2 N A cos α r 2 2 π r ( sin α ) r d α = π N A .
P = I 0 A T ,
J 0 = I 0 A T / π
I = J 0 / r 2 = I 0 A T / π r 2 .
I g = I 0 ( W s L ) T D T c / π r 2 ( 1 - b r n - 1 n ) 2 .
τ = 16 n 2 T D 3 ( 1.045 ) h T c T s I s I g .
A = F G w / G g = I w / I g .
I s I g = I s I w I w I g = I s I w a T s a T c a ,
τ = 16 n 2 T D 3 ( 1.045 ) h R w R c a I s I w ,
R w / R c = ( T s a / T s ) / ( T c a / T c ) .
I R = I 0 A R cos i / π r 2 ,
I / I R = T / 0.707 R .
( G R / G 0 ) ( T / 0.707 R ) = 1.