An Analogy Between Solutions of Electrostatic and Gravitational Nature in Hollow Spheres

1

An Analogy between solutions of electrostatic and gravitational nature in hollow spheres

by

Pratibha Chopra-Sukumaran

A project submitted in partial fulfillment of the requirements of the degree of

Master’s in Education

State University of New York College at Buffalo

2009

Approved by

Program Authorized to Offer Degree

Date

State University of New York College at Buffalo

An Analogy between solutions of electrostatic and gravitational nature in

Hollow Conducting Sphere

by Pratibha Chopra-Sukumaran

Supervisory Faculty: Dr. Dan MacIsaac, Associate Professor, Physics Department

Dr. Luanna Gomez, Assistant Professor, Physics Department

Abstract

The purpose of this paper is to relate the solutions of a conducting shell with uniform charge density to that of hollow earth of uniform mass density. Both problems involve the use of vector superposition within the empty space of a shell. The mathematical form, as a function of densities, positions and directions in which they act, allows us to form similar solutions inside and outside their respective shells. General solutions for the two problems for which points are chosen off-center will be described. The importance of these solutions is to demonstrate and to motivate the use of quasi-qualitative use of vector superposition in the physics classroom.

Introduction

In this paper, the effect of charges on a conducting surface of a hollow sphere of radius Rc is compared to that of a hollow earth whose total mass is uniformly distributed on a spherical shell of the radius equal to the earth’s mean radius Re The solutions to the net forces inside the shells will be reproduced from several physics teaching resources.1, 2, 3 Coulomb’s law and Gauss’s law within a spherical conducting surface are presented for both the systems.

The analogy between a hollow conducting sphere and a hollow earth arises from the nature of force as function of positions between points on the sphere and a position inside or outside the sphere.1, 2 The earth is typically modeled as a solid sphere of uniform volume and mass density. Inside the respective spheres, Coulomb’s, Gauss’s4 and Universal Gravitation laws will be applied.1 Outside the spheres, the concepts of an electric and gravitation flux passing through a closed surface (a sphere of radius R), and the electric and gravitational forces outside the sphere will be introduced.5

Coulomb’s Law Gauss’s Law within a Spherical Conducting Surface

When a charge is enclosed in a uniform shell as shown in Fig. 1, the net field inside the

shell is zero.1 This can be argued with the use of Coulomb’s law and Gauss’s law. According to Coulomb’s law, the electric force due to charges and at a distance falls off rapidly as. The equation for electric force is

(1)

where, is the distance in meters between the charges, is the unit vector and is the permittivity of free space. In SI unitsis equal to 8.8542 x 10-12 C2N-1m-2 .6

Consider an enclosed point charge B near the inside surface of a charged hollow sphere (Fig. 1.)1 Imagine that the point charge is so small that it does not alter the distribution of charges placed on the outside of the sphere itself. The charges to the upper left of the location B on the surface are closer to B. Therefore, each force exerts a greater force to pull the charge towards the surface. However, there are many more charges farther away from B and this result in electric field pointing to the right and downwards towards the surface.

Fig. 1

Gauss’s law for electric fields relates the spatial behavior of the electrostatic field with the charge distribution that produces it, and relates patterns of electric field over a closed Gaussian surface to the amount of charge inside the closed surface. To calculate the electric field from Gauss’s law, symmetry and superposition of the electric field are required. Thein Gauss’s law represents the total electric field at each point on the surface.

(2)

is a vector with magnitude directly proportional to force on a small charge , with units of Newtons per coulomb (N/C). To develop the concept of flux, the concept of electric field needs to be clearly explained to students. According to Gauss’s law, electric charge produces an electric field, and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface. The flux of a field through a curved surface is described by a surface integral. The integral form for flux taken from Maxwell’s equations3, 4 is given in eq. 3 below,

(3)

where is the electric field, is the unit vector (outward) normal to the surface, and is the charge in coulombs. The left side of equation 3 is the summation of the electric fluxes through a vast number of very tiny pieces of a surface. This electric flux can be thought of as the number of electric field lines passing through a closed surface. The right side of the equation is the total amount of charge contained within that surface divided by the permittivity of free space. For a sphere with surface area of, the electric field as according to Coulomb’s law is. Then the electric flux through the spherical surface is

(4)

Gauss’s law is completely equivalent to Coulomb’s law for static charges. Coulomb’s law is not valid for moving charges. In Gauss’s law, electric field represents the total electric field at each point on the surface under consideration. The magnitude of electric field at all locations on the conducting surface is given by Coulomb’s law. The electric field is not absorbed by space and can come out from a positive charge and end at a negative charge. This is analogous to flow of water from a sink to a drain. The charge is conserved and not lost on the way!!

High School Learning

Gauss’s law is a difficult concept for students to understand. Both Gauss’s law and Coulomb’s law can be understood better through demonstrations. There are several demonstrations for Gauss’s law and Coulomb’s law which can provide a clearer understanding of the concepts for students studying physics in high school.7, 8

One of the demonstrations to show that charge inside a closed conducting surface is zero uses the Faraday’s cage. The demonstration requires a wire screen Faraday cage with pith balls and charging materials. When the cage is charged, the aluminum foil "pith balls" move away from the outside of the cage, indicating the presence of an electric field. However, on the inside of the cage the pith balls do not move from the wire cage, indicating that there is no charge and therefore no electric field on the inside of the conducting surface. This demonstration and others can be viewed on the website.7 Coulomb’s law can be demonstrated7, 9 using a torsion balance showing that the electrostatic force varies as the square of the distance and is proportional to the charge. Students can observe several demonstrations and simulations of Gauss’s and Coulomb’s laws which involve the use of the Van de Graff generator.10

Gravitational Field due to Hollow Earth with uniformly Distributed Mass

Consider hollow earth represented above, with a mass near the inside surface of the

hollow sphere (Fig. 2). The gravitational force between the hollow earth of mass and point mass is

(5)

Figure 2

where, is the distance in meters between the masses, is the unit vector and G is the Gravitational constant, with a value of in SI units. The negative sign indicates that the force is attractive. The magnitude of gravitational force is.2

(6)

The point mass is so small that it does not alter the mass distribution and density of the hollow sphere. Each point on the hollow earth nearer to the point inside exerts a greater force to pull the mass towards the surface. However, there is a greater surface area of the hollow sphere farther away towards left and downwards from the point and results inpointing downwards towards the surface. The greater mass below balances the greater force per unit charge. The gravitational field on test mass, (a mass much smaller than that of the earth very near the earth’s surface) is

(7)

is defined as the gravitational force per unit mass, labeled as with a value of or in more familiar terms it is the acceleration due to gravity. Since the gravitational field falls off rapidly as this results in the forces to the right and left canceling each other out, leading to a net zero field inside the sphere.

This has also been proved mathematically by Crowell.2 Without going into mathematical derivation, it can be shown that the gravitational field of a uniform hollow earth of mass interacting with point like mass outside the hollow earth is

(8)

where is the center to center distance. If mass is inside the shell, then the energy is constant, i.e. the shell’s interior gravitational field is zero, given by

(9)

for mass inside the hollow earth sphere and is the distance from the center of hollow earth.

The alteration of the space around a mass is called gravitational field and space around a charge changes to create an electric field. There is similarity between the gravitational energy of a uniform spherical shell of mass, interacting with a point like mass outside it and electric field due to a point charge inside and outside a hollow sphere of charge. The gravitational energy equals, and is the gravitational constant. In Gauss’s law for electrostatics, is replaced by, which equals. The value of in SI units is .5 If mass is inside the shell, then the energy is constant, in other words, the gravitational field in the shell’s interior is zero. The electrostatic energy on the interior of hollow shell is also zero. By knowing the electric field due to charges on the outer surface, and expressing the electric field on a surface quantitatively in terms of electric flux, the charge inside the closed surface can be obtained.

The gravitational energy and electrostatic energy both follow a 1/r dependence, as shown in the graph below graph from Crowell.2

Conclusion

Gauss’s law is useful in calculating the electric flux knowing the electric field over a surface. Gauss’s law in essence is a physical analogy between flux and electric field. However, electric flux should not be confused as a physical movement of particles. Using the concept of flux, it is easier for students to understand that the charge inside a closed surface will always be zero. The integral form of Gauss’s law for electric fields can be understood by explaining each term in equation 3. It shows that any charge located on the outside surface produces an equal amount of inward flux and outward flux and that the net contribution the flux through the surface must be zero. There is an analogy between the gravitational energy for a uniform spherical shell interacting with a point like mass inside it, and electric field due to interaction of a point charge inside and outside a hollow sphere of charge. Equations 7-9 illustrate this analogy. Classroom demonstrations and simulations can aid students understand these difficult concepts from fundamental principles.

Acknowledgement

This manuscript fulfills the requirements for PHY 690 at SUNY – Buffalo State College. Discussions, suggestions and help in the preparation of this manuscript from Drs. MacIsaac and Gomez are gratefully acknowledged. Author also acknowledges support from Noyce Fellowship for pursuing the M. S. Ed. program and NSF for attending Summer Workshops and Teacher Training Fellowship.

References

1.  R. W. Chabay, B. A. Sherwood, Electric and Magnetic Interactions (Wiley,

Hoboken, NJ, 1995), p. 14- 15, p. 146, p. 404.

2.  B. Crowell, Simple Nature (2008), p. 101-102 http://www.lightandmatter.com/html_books

3.  R. Knight, Physics for Scientists and Engineers with Modern Physics: A Strategic Approach (Pearson Education, Inc., Addison Wesley, San Francisco, CA, 2004).

4.  D. Fleisch, A Student’s Guide to Maxwell’s Equations (Cambridge University Press,

NY, 2008).

5.  S. Saeli, & D. MacIsaac, “Using Gravitational Analogies to introduce Elementary Electrical Field Theory Concepts,” Phys. Teach., 45, 104 (2007)

6.  D. A. McQuarrie, J. D. Simon, Physical Chemistry A Molecular Approach (University Science Books, Sausalito, CA, 2002), p18

7.  http://www.physics.ucla.edu/demoweb/demomanual/electricity_and_magnetism/electrostatics/gausss_law.html

8.  http://www.phys.uvic.ca/resman/electricity_and_magnetism.html#5B1

9.  http://www3.interscience.wiley.com:8100/legacy/college/halliday/0471320005/simulations6e/index.htm).

10.  http://www.virlab.virginia.edu/VL

BIOGRAPHY

Pratibha Chopra-Sukumaran has a B.S. and M.S. in chemistry from India. She completed her doctorate in 1990 from SUNYAB. She has served as a lecturer at SUNY Empire State College for ten years and as a Visiting Lecturer at SUNY College at Buffalo for three years. She currently resides in Amherst with her husband and daughter. She can be contacted at