Poisson’s Equation Section 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: Φ m  ∫ S n  g da = “amount of g passing through surface S” n  Unit vector normal to S at differential area da.

Φ m  ∫ S n  g da Use g = -G(m/r 2 ) e r n  e r = cosθ  n  g = -Gm(r -2 cosθ) So: Φ m = -Gm ∫ S (r -2 cosθ)da da = r 2 sinθdθdφ  ∫ S (r -2 cosθ)da = 4π  Φ m = -4πGm (ARBITRARY S!)

We’ve just shown that the Gravitational Flux passing through an ARBITRARY SURFACE S surrounding a mass m (anywhere inside!) is: Φ m = ∫ 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!)

Φ m = ∫ S n  g da = - 4πGm Gauss’s Law for Gravitation Generalizations: Many masses in S: –Discrete, point masses: m = ∑ i m i  Φ m = - 4πG ∑ i m i = - 4πG M enclosed where M enclosed  Total Mass enclosed by S. –A continuous mass distribution of density ρ: m = ∫ V ρdv (V = volume enclosed by S)  Φ m = - 4πG∫ V ρdv = - 4πG M enclosed (1) where M enclosed  ∫ 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!!

For a continuous mass distribution: Φ m = - 4πG∫ V ρdv (1) –But, also Φ m = ∫ S n  g da = - 4πG M enclosed (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 = (ρ c /ε)

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   2 Ф = 4πGρ Poisson’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,  2 Ф = 0 Laplace’s Equation!

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 density of 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.

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(x 1,x 2,x 3 ) = 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 If Φ = constant, g (obviously!) = 0  g has no component along an equipotential surface!

Gravitational Field g  -  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.  The field 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.

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

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 Ф is a 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.

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