Transcript Sects. 5.2 (II)

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: Φm  ∫S ng da
= “amount of g passing
through surface S”
n  Unit vector normal to S at differential area da.
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 (ARBITRARY S!)
•
We’ve just shown that the Gravitational Flux
passing through an ARBITRARY SURFACE S
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!)
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)
Note!! where Menclosed  ∫V ρdv  Total Mass enclosed by S.
This is    If S is highly symmetric, we can use (1) to
calculate the gravitational field g! Examples next!
important!!
• 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)
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 = 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!)
2Ф = 0
Laplace’s Equation!
• If ρ = 0 in the region where we want Φ,
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(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
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 r1 & r2 from 2 masses.
Equipotential surface is: Φ = -GM[(r1)-1 + (r2)-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/(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 )