On the Gravitational Field of a Sphere of Charge

G.R.Dixon, June 24, 2009

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Reference: The Feynman Lectures on Physics, V2, Chapter 8.

Abstract. Building on ideas presented by Feynman, a formula for the gravitational field of a solid sphere of charge is developed.

In Sect. 8-5 of the Reference, Feynman discusses the locality of electrostatic field energy. He states, "There is also a physical reason why it is imperative that we be able to say where energy is located. According to the theory of gravitation, all mass is a source of gravitational attraction. We also know, by E=mc2, that mass and energy are equivalent. All energy is, therefore, a source of gravitational force."

Feynman later develops the formula for u, the energy density of the field:

. (1)

Fig. 1 depicts a solid sphere of charge Q=-1.6E-19 coul (the electron charge) and radius R=2.81794E-15 meters (the classical electron radius). We wish to develop a formula for the gravitational field, g(r), attributable to the electric energy inside the sphere of radius r>R.

Figure 1

Q and Associated Gravitational Field

As Feynman notes in Eq. (8.7), the total field energy of Q (in all of space) is

. (2)

(This is equivalent to a field mass of 5.44583E-31 kg ... somewhat less than the commonly accepted electron mass of 9.11E-31 kg.) The field energy within the sphere of radius r is then the total field energy minus the field energy outside the sphere:

. (3)

Now

. (4)

Thus

. (5)

Invoking E=mc2 and Newton’s Law of Gravitation,

. (6)

Fig. 2 plots the computed electric field energy’s gravitational field vs. r over the range R<r<5R. Note the effect of the 1/r3 term up close to Q. Fig. 3 plots g vs. r using the "mechanical mass" for the electron. (In this article the mechanical mass is defined to be the standard mass of 9.11E-31 kg minus the electric field energy’s equivalent mass of 5.44583E-31 kg). Fig. 4 plots the combined field energy plus mechanical mass field, and the g field of the standard electron mass of 9.22E-31 kg. Note how, at small values of r, the standard electron mass g field dominates. But at larger r the two converge to a common function of r.

Figure 2

 

Field Energy Gravitational Field

Figure 3

 

Mechanical Mass Gravitational Field

Figure 4

Standard g Field and Combined Fld Energy and Mech Mass Field

 

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