Poisson’s Equation Section 5.2 (“Fish’s” Equation!)

2. Poisson’s EquationSection 5.2(“Fish’s” Equation!) • Comparison of properties of gravitational fields with similar properties of electrostatic fields (Maxwell’s equations!) • Consider an arbitrary surface S, as in the figure. A point mass m is placed inside. • Define: Gravitational Flux through S: Φm  ∫S ng da = “amount of g passing through surface S” n Unit vector normal to S at differential area da.

3. Φm  ∫S ng da Use g = -G(m/r2) er ner = cosθ  ng = -Gm(r-2cosθ) So: Φm = -Gm ∫S (r-2cosθ)da da = r2sinθdθdφ ∫S (r-2cosθ)da = 4π  Φm= -4πGm (ARBITRARYS!)

4. We’ve just shown that the Gravitational Flux passing through an ARBITRARY SURFACES surrounding a mass m (anywhere inside!) is: Φm = ∫S ng da = - 4πGm (1) (1) should remind you of Gauss’s Law for the electric flux passing though an arbitrary surface surrounding a charge q (the mathematics is identical!). (1) = Gauss’s Law for Gravitation (Gauss’s Law, Integral form!)

5. Φm = ∫S ng da = - 4πGm Gauss’s Law for Gravitation • Generalizations: Many masses in S: • Discrete, point masses: m = ∑i mi Φm = - 4πG ∑i mi = - 4πG Menclosed where Menclosed  Total Mass enclosed by S. • A continuous mass distribution of density ρ: m = ∫V ρdv (V = volume enclosed by S) Φm = - 4πG∫V ρdv = - 4πG Menclosed (1) where Menclosed  ∫V ρdv Total Mass enclosed by S. If S is highly symmetric, we can use (1) to calculate the gravitational field g! Examples next! Note!! This is    important!!

6. For a continuous mass distribution: Φm = - 4πG∫V ρdv(1) • But, also Φm = ∫S ng da = - 4πG Menclosed(2) • The Divergence Theorem from vector calculus (Ch. 1, p. 42): (Physicists correctly call it Gauss’s Theorem!): ∫S ng da  ∫V (g)dv (3) (1), (2), (3) together:  4πG∫V ρ dv = ∫V (g)dv surface S & volume V are arbitrary  integrands are equal!  g= -4πGρ (Gauss’s Law for Gravitation, differential form!) Should remind you of Gauss’s Law of electrostatics: E= (ρc/ε)

7. Poisson’s (“Fish’s”) Equation! • Start with Gauss’s Law for gravitation, differential form: g= -4πGρ • Use the definition of the gravitational potential:g  -Φ • Combine:(Φ) = 4πGρ  2Ф = 4πGρPoisson’s Equation! (“Fish’s” equation!) • Poisson’s Equation is useful for finding the potential Φ(in boundary value problems similar to those in electrostatics!) • If ρ = 0 in the region where we want Φ,2Ф = 0 Laplace’s Equation!

8. Lines of Force & Equipotential Surfaces Sect. 5.3 • Lines of Force(analogous to lines of force in electrostatics!) • A mass M produces a gravitational field g. Draw lines outward from M such that their direction at every point is the same as that of g. These lines extend from the surface of M to  Lines of Force • Draw similar lines from every small part of the surface area of M: These give the direction of the field g at any arbitrary point. • Also, by convention, the densityof the lines of force (the # of lines passing through a unit area  to the lines) is proportional to the magnitude of the force F (the field g) at that point.  A lines of force picture is a convenient means to visualize the vector property of the g field.

9. Equipotential Surfaces • The gravitational potential Φ is defined at every point in space (except at the position of a point mass!).  An equation Φ = Φ(x1,x2,x3) = constant defines a surface in 3d on which Φ = constant(duh!) • Equipotential Surface: Any surface on which Φ = constant • The gravitational field is defined as g  - Φ IfΦ = constant, g (obviously!) = 0  g has no component along an equipotential surface!

10. Gravitational Fieldg  - Φ  g has no component along an equipotential surface.  The force F has no component along an equipotential surface.  Every line of force must be normal () to every equipotential surface.  Thefield g does no work on a mass m moving along an equipotential surface. • The gravitational potential Φ is a single valued function.  No 2 equipotential surfaces can touch or intersect. • Equipotential surfaces for a single, point mass or for any mass with a spherically symmetric distribution are obviously spherical.

11. Consider 2 equal point masses, M, separated, as in the figure.Consider the potential at point P, a distances r1 & r2 from 2 masses.Equipotential surface is: Φ = -GM[(r1)-1 + (r2)-1] = constant • Equipotential surfaces look like this 

12. When is the Potential Concept Useful? Sect. 5.4 • A discussion which (again!) borders on philosophy! • As in E&M, the potential Ф in gravitation is a useful & powerful concept / technique! • Its use in some sense is really a mathematical convenience to the calculate the force on a body or the energy of a body. • The authors state that force & energy are physically meaningful quantities, but that Ф is not. • I (mildly) disagree. DIFFERENCES in Ф are physically meaningful! • The main advantage of the potential method is that Фisa scalar (easier to deal with than a vector!). • We make a decision about whether to use the force (field) method or or the potential method in a calculation on case by case basis.

13. Example 5.4Worked on the board! • Consider a thin, uniform disk, mass M, radius a. Density ρ =M/(πa2). Find the force on a point mass m on the axis. • Results, both by the potential method & by direct force calculation: Fz = 2πρG[z(a2 + z2)-½ - 1] (<0 )