Section 13–4 Gravitational field of large objects

(Infinite plane sheet / Thin spherical shell / Inside a sphere)

In this section, Feynman discusses how a gravitational field or potential energy varies due to an infinite plane sheet, thin spherical shell, and inside a sphere.

1. Infinite plane sheet:

“The gravitational field C at a mass point produced by an infinite plane sheet of matter (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

Feynman illustrates the validity of Gauss’ law for gravitation by calculating the gravitational field produced by an infinite plane sheet of material. He explains that the gravitational force is independent of distance because if we are closer, most of the matter is pulling at an unfavorable angle; if we are farther away, even more matter is situated more favorably to exert a pull toward the plane. Specifically, the force is weaker by the inverse square if we are farther away, but within the same cone of solid angle, it would be compensated by more matter that is proportional to the square of the distance. In other words, the farther away an object is from the infinite plane, more mass will be included within a “circle of influence” (or field of view with a cone of solid angle). Physics teachers could emphasize that Gauss’ law is applicable here because the gravitational force follows the inverse square law.

Gauss’ law for gravitation can be stated as: the flux of gravitational field through any surface is proportional to the total gravitational charge (mass). Similarly, we can use Gauss’ law to solve an electrical problem involving a large electrically charged plate that has an amount σ of charge per unit area. The electric field at any point outside the large plate is approximately equal to σ/2ϵ₀ in the outward direction and the approximation gets better if the plate is larger. If the plate is infinitely large, any point at a distance away from the plate is equivalent to every other point away from the plate. In Feynman Lectures on Computation, he adds that “defining a charge density here is important — it would be meaningless to discuss the total charge for an infinite plate (Feynman, 1996, p. 232).” This is because the total charge would be infinity in the idealized infinite plate.

2. Thin spherical shell:

“Thus, for a thin spherical shell, the potential energy of a mass m′, external to the shell, is the same as though the mass of the shell were concentrated at its center (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

We have assumed the gravitational force produced by the Earth at a point on the surface or outside it, is the same as if all the mass of the Earth were located at its center. This assumption is related to Newton’s shell theorem in which we conceptualize a spherical body as thin uniform hollow shells. Feynman’s approach is to work with the potential energy instead of the gravitational force because we do not have to worry about angles, that is, we merely add the potential energies of all the particles. However, one may expect Feynman to take the derivative of the potential energy (or at least mention the result) to obtain the gravitational force. This step is not really complicated and it may help students to appreciate Newton’s shell theorem better.

It may not be clear to students how the work done results in the potential energy due to a ring of mass is −Gm′dm/r. In a sense, Feynman partially fills up this gap only in the next chapter. In Chapter 14, he adds that “[t]he potential energy of gravitation for point masses M and m, a distance r apart is U(r) = −GMm/r (14.5). The constant has been chosen here so that the potential is zero at infinity (Feynman et al., 1963, section 14–3 Conservative forces).” Importantly, the work done by gravity results in negative gravitational potential energy because the gravitational force (not external force) is attractive and brings objects closer. The decrease in potential energy also implies an increase in the kinetic energy of the objects.

Lastly, Feynman only mentions the Earth acts as if all the materials were at the center by imagining it as a series of spherical shells. It is not trivial to explain how the gravitational force between two spherical objects is equivalent to the case when the mass of both objects are located at their center of mass. Young and Freedman (2004) explain that “the forces the two bodies exert on each other are an action-reaction pair, and they obey Newton’s third law. So we have also proved that the force that m exerts on the sphere M is the same as though M were a point. But now if we replace m with a spherically symmetric mass distribution centered at m’s location, the resulting gravitational force on any part of M is the same as before, and so is the total force. This completes our proof.” Notably, they have assumed that the distribution of mass remains spherical (or undistorted) in the presence of a massive object.

3. Inside the sphere:

“If the potential energy is the same no matter where an object is placed inside the sphere, there can be no force on it (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

Inside a spherical shell, the work done on an object by gravity, W = −Gm′m/a, is independent of R and independent of position. Therefore, no force is needed to move the object anywhere within the hollow sphere and no work is done when we move it about inside. It is similar to no work is done in moving the object horizontally on a flat table. (In short, W = −Gm′m/a Þ same potential energy within a shell Þ gravitational force, F = 0 within a shell.) If a point mass m is positioned at a distance r from the center of the Earth, we can determine the gravitational force by ignoring the spherical shells whose radius are longer than r; we only need to focus on the spherical shells whose radius are shorter than r.

Note: Physics teachers could compare the potential energy inside and outside a shell as shown below.

(i) Outside the shell: r_max – r_min = (R + a) – (R – a) = 2a

 Þ W = −Gm′m(r_max – r_min)/2aR = −Gm′m/R

(ii) Inside the shell: r_max – r_min = (R + a) – (a – R) = 2R

 Þ W = −Gm′m(r_max – r_min)/2aR = −Gm′m/a (constant)

The validity of Gauss’ law depends on the inverse square law of Coulomb. In volume II, Feynman provides a better explanation: “[i]f the force law were not exactly the inverse square, it would not be true that the field inside a uniformly charged sphere would be exactly zero. For instance, if the force varied more rapidly, like, say, the inverse cube of r, that portion of the surface which is nearer to an interior point would produce a field which is larger than that which is farther away, resulting in a radial inward field for a positive surface charge (Feynman et al., 1964, section 5.8 Is the field of a point charge exactly 1/r²?).” Thus, the net electric field is also zero for any two symmetric or opposite cones (same solid angle) diverging from the opposite sides of a point P as the fields due to them are in opposite direction (See Feynman et al., 1964, Fig. 5–9). We can apply the same logic for gravitational fields.

Questions for discussion:

1. How would you explain the gravitational field produced by an infinite plane sheet of material is constant or independent of distance?

2. Why is the gravitational force due to the Earth at a point on the surface or outside it, is the same as if all the mass of the earth were located at its center?

3. Why is the gravitational force equal to zero for any point within a spherical shell of mass?

The moral of the lesson: the gravitational force due to the Earth at a point on its surface or beyond, is the same as if all the mass of the Earth were located at its center, can be shown using Newton’s shell theorem.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

3. Feynman, R. P. (1996). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley.

4. Young, H. D., & Freedman, R. A. (2004). Sears and Zemansky’s University Physics (11th ed.). California: Addison Wesley.