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A second example is a charged particle that moves in a synchrotron. In the absence of

external elds, such a particle will orbit forever in a circular orbit without any change in

its speed. In reality, a charged particle is coupled to electromagnetic elds. This has the

e ect that a charged particle that is accelerated emits electromagnetic radiation, called

synchrotron radiation 31]. The radiated energy means an energy loss of the particle, so

that the particle slows down. This is actually the reason why accelerators such as used

at CERN or Fermilab are so large. The acceleration of a particle in a circular orbit with

radius r at the given velocity v is given by v2 =r. This means that for a xed velocity v

the larger the radius of the orbit is, the smaller the acceleration is, and the weaker the

energy loss due to the emission of synchrotron radiation is. This is why one needs huge

machines to accelerate tiny particles to an extreme energy.

Problem i: The modes in the plate in gure 16.1 are also damped because of radiation

damping. What form of radiation is emitted by this oscillating plate?

Chapter 17

Potential theory

Potential elds play an important role in physics and geophysics because they describe the

behavior of gravitational and electric elds as well as a number of other elds. Conversely,

measurements of potential elds provide important information about the internal struc-

ture of bodies. For example, measurements of the electric potential at the Earth's surface

when a current is send in the Earth gives information about the electrical conductivity

while measurements of the Earth's gravity eld or geoid provide information about the

mass distribution within the Earth.

An example of this can be seen in gure 17.1 where the gravity anomaly over the

northern part of the Yucatan peninsula in Mexico is shown 29]. The coast is visible as a

thin white line. Note the clear ring-structure that is visible in the gravity signal. These

rings have led to the discovery of the Chicxulub crater that was caused by the massive

impact of a meteorite. Note that the diameter of the impact crater is about 150 km!

This crater is presently hidden by thick layers of sediments, at the surface the only visible

imprint of this crater is the presence of underground water- lled caves called \cenotes"

at the outer edge of the crater. It is the measurement of the gravity eld that made it

possibly to nd this massive impact crater.

The equation that the gravitational or electrical potential satis es depends critically

on the Laplacian of the potential. As shown in section 4.5 the gravitational eld has the

mass density as its source:

(r g) = G (4:17) again

;4

The gravity eld g is (minus) the gradient of the gravitational potential: g = . This

;rV

means that the gravitational potential satis es the following partial di erential equation:

2 V (r) = 4 G : (17.1)

r

This equation is called the Laplace equation, it is the prototype of the equations that

occur in potential eld theory. Note that the mathematical structure of the equations of

the gravitational eld and the electrical eld is identical (compare the equations (4.12)

and (4.17)), therefore the results derived in this chapter for the gravitational eld can be

used directly for the electrical eld as well by replacing the mass density by the charge

density and by making the following replacement:

|{z} | {z 0 :

4G (17.2)

}

, ;1="

Gravity Electrostatics

241

CHAPTER 17. POTENTIAL THEORY

242

91Ëš 90Ëš 89Ëš 88Ëš

22Ëš 22Ëš

21Ëš 21Ëš

20Ëš 20Ëš

91Ëš 90Ëš 89Ëš 88Ëš

Figure 17.1: Gravity eld over the Chicxulub impact crater on the northern coast of

Yucatan (Mexico). The coastline is shown by a white line. The numbers along the vertical

and horozontal axes refer to the latitutde and longitude respectively. The magnitude of

the horizontal gradient of the Bouguer gravity anomaly is shown, details can be found in

Ref. 29]. Courtesy of M. Pilkington and A.R. Hildebrand.

The theory of potentials elds is treated in great detail by Blakeley 10].

17.1 The Green's function of the gravitational potential

The Laplace equation (17.1) can be solved using a Green's function technique. In essence

the derivation of the Green's function yields the well-known result that the gravitational

potential for a point-mass m is given by The use of Green's functions is introduced

in great detail in chapter 14. The Green's function G(r r0 ) be the potential at location r

;Gm=r.

generated by a point mass at location r0 satis es the following di erential equation:

r0 ) = ;r r0 :

2 G(r (17.3)

r ;

Take care not to confuse the Green's function G(r r0 ) with the gravitational constant G.

Problem a: Show that the solution of (17.1) is given by:

Z

V (r) =4 G G(r r0 ) (r0 )dV 0 : (17.4)

17.2. UPWARD CONTINUATION IN A FLAT GEOMETRY 243

Problem b: The di erential equation (17.3) has translational invariance, and is invariant

for rotations. Show that this implies that G(r r0 ) = G(jr r0 Show by placing

the point mass in the origin by setting r0 = 0 that G(r) satis es

; j).

2 G(r) = (r) : (17.5)

r

Problem c: Express the Laplacian in spherical coordinates and show that for r > 0

equation (17.5) is given by

1 @ r2 @G(r) = 0 : (17.6)

r2 @r @r

Problem d: Integrate this equation with respect to r to derive that the solution is given

by G(r) = A=r + Br, where A and B are integration constants.

The constant B must be zero because the potential must remain nite as r The

! 1.

potential is therefore given by

G(r) = A : (17.7)

r ;

Problem e: The constant A can be found by integrating (17.5) over a sphere with radius

R

R centered around the origin. Show that Gauss' theorem implies that 2G(r)dV =

H r

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