Guide 20-3. A Comparative Guide to Gravitational and Electric Forces, Fields, and Energy

In order to understand the concepts of electric force and field, it may help to make analogies to gravitational forces and fields.

Summary of important concepts and relationships for gravitational and electrical forces and fields
Row Attribute Gravitational Electrical Condition for validity Notes
0 Force Law Newton's Law of Universal Gravitation:
Fg = Gm1m2/r²		 Coulomb's Law:
|Fel| = k|q1||q2|/r²		 point masses or charges or uniform spherical distributions of mass or charge The force laws have the same mathematical form. They depend on distance in the same way. Gravitational force depends on the product of the masses, and electrical force depends on the product of the charges. Gravitational force is always attractive, while electrical force can be attractive or repulsive.
1 Source of field Collection of masses--This could be a single point mass or a distribution of masses. Collection of charges--This could be a single point charge or a distribution of charges.  

For both gravitational and electric fields, the field is created by a particle or distribution of particles. For electric fields, the particle(s) creating the field are charged particles. The magnitude and direction of the electric field at any particular point in space is determined with a positive test charge. The test particle is assumed to have a small enough charge that it has negligible influence on the field. The field is calculated as the ratio of the electric force on the test charge to the value of the charge. Conversely, if you place a charge q in the field, the electric force on the charge is the product of q and the field.

For gravitational fields, the particle(s) creating the field are masses. The magnitude and direction of the gravitational field at any particular point in space is determined with a test particle. The test particle is assumed to have a small enough mass that it has negligible influence on the field. The field is calculated as the ratio of the gravitational force on the test mass to the value of the mass. Conversely, if you place a mass m in the field, the gravitational force on the charge is the product of m and the field.

Note that E = Fel/q0 and Felec = qE are vector equations, and q and q0 are written without absolute value marks. This means that if the charge is positive (negative), the force is in the same (opposite) direction as the field.
2 Test particle A mass that is small enough to have negligible influence on the field A charge that is small enough to have negligible influence on the field. By convention, the test charge is positive.  
3 Magnitude and direction of field g = Fg/m, where m is the value of the test mass.
This is a vector relationship. The vector field g is in the same direction as the force on the test mass.		 E = Fel/q0, where q0 is the test charge (both magnitude and sign).
This is a vector relationship. The vector field E is in the same direction as the force on a positive test charge.		 As definitions, these are always valid.
4 Force on a particle in the field Fg = mg, where m is the mass of the particle. Fel = qE, where q is the charge of the particle. always valid
5 Magnitude of the field of a point particle g = GM/r², where M is the mass of the particle producing the field E = k|Q|/r², where Q is the charge of the particle producing the field only valid for point charges The definitions of field given in Row 3 are operational definitions in that one determines the field by the operation of putting a test particle in the field and measuring the force--whether electric or gravitational--on the particle. There are other ways to determine fields using known physical laws. For point particles, we use Newton's Law of Gravitation for masses and Coulomb's Law for charges. Those are given in this row. We emphasize that these are point particles.
6 Field of a distribution of point particles Vector sum of the gravitational forces on the individual particles Vector sum of the electrical forces on the individual particles always valid If one has a distribution (collection) of point particles, the net field is the superposition of the individual fields of all the particles. If there are only a few particles, the net field is relatively easy to calculate. For large collections of particles, doing a vector addition of the individual fields of all the point particles would be prohibitive. However, there are mathematical ways to calculate such fields if the distribution of mass or charge has a symmetric shape such as a sphere, cylinder, or plane. There is some discussion of these methods in Section 19-7 of the text. This is optional reading. You are, however, expected to be able to use the field formulas for certain distributions that occur frequently. For both gravitational and electric fields, one such distribution is a uniform spherical distribution of mass or charge.
7 Field of a uniform spherical distribution (for points outside the sphere) g = GM/r², where M is the mass of the sphere and r is the distance from the center of the sphere to the point where the field is determined. E = k|Q|/r², where Q is the charge of the sphere and r is the distance from the center of the sphere to the point where the field is determined. valid for a uniform spherical distribution
8 Potential energy of a system of two point particles Ug = -Gm1m2/r, where the gravitational potential energy is defined to be 0 at infinity Uel = kq1q2/r, where the electric potential energy is defined to be 0 at infinity only valid for point charges The analogy between electric and gravitational forces carries through to the concept of potential energy with one important difference. The gravitational potential energy of a system of two point masses is negative while the electric potential energy of two positive (or two negative) point charges is positive. This is a consequence of the fact that the masses attract while the charges repel. Consider this argument for a system of two masses.  If the masses move further apart, they do so against the gravitational field, and the gravitational potential energy of the system increases. This agrees with the formula in this row, because as r increases, Ug = -Gm1m2/r becomes less negative. Now consider the situation for the two like charges. If the charges move closer together, they do so against the electric field, and the electric potential energy of the system increases. This agrees with the formula in this row, because as r decreases, Uel = kq1q2/r becomes mores positive.  Of course, if the charges are opposite in sign, Uel will be negative, since the product q1q2 will be negative. This is analogous to the situation with attracting masses for gravitation.
9 Change in potential energy for uniform field ΔUg = mgΔy, where Δy is the change in height of mass m. ΔUel = -qEΔs, where Δs is the displacement of charge q in the direction of the field. only valid for uniform fields Many problems deal with situations where the field is constant or nearly so. In that case, the formulas for potential energy reduce to linear forms. We have seen that for objects near the surface of the Earth, the vertical distances moved are much smaller than the radius of the Earth. In that case, the change in gravitational potential energy is calculated using the familiar formula in this row. For uniform electric fields, which can be produced by two equal and opposite uniform distributions of charge on parallel plates, there is an analogous formula as shown in the table. Note that an increase in height gives a corresponding increase in gravitational potential energy. However, a displacement in the direction of an electric field gives a decrease in electric potential energy. In the former case, the mass moves against the field and in the latter case the charge moves with the field. This is the reason for the difference in signs. If, however, we define a quantity Δyel = -Δs as an electrical height analogous to gravitational height, Δy, then ΔUel = qEΔyel has the same sign as ΔUg = mgΔy. In this formulation, we interpret Δyel as the displacement of the particle opposite the electric field direction in the same way that Δy is the displacement opposite the gravitational field direction.

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