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 ng da

= “amount of g passing

through surface S”

n Unit vector normal to S at differential area da.

Φm ∫S ng da

Use g = -G(m/r2) er

ner = cosθ

ng = -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 ng 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 ng 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!!

• For a continuous mass distribution:

Φm = - 4πG∫V ρdv (1)

– But, also Φm = ∫S ng da = - 4πG Menclosed (2)

– The Divergence Theorem from vector

calculus (Ch. 1, p. 42): (Physicists correctly

call it Gauss’s Theorem!): ∫S ng 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!)

• If ρ = 0 in the region where we want Φ, 2Ф = 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(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 )