potential of hollow ball

potential of hollow ball

Let

(

ξ

,

η

,

ζ

)

be a point bearing a mass

m

and

(

x

,

y

,

z

)

a point. If the

distance

of these points is

r

, we can define the

potential

of

(

ξ

,

η

,

ζ

)

in

(

x

,

y

,

z

)

as

m

r

=

m

(

x

-

ξ

)

2

+

(

y

-

η

)

2

+

(

z

-

ζ

)

2

.

The relevance of this concept appears from the fact that its

partial derivatives

∂

∂

⁡

x

⁢

(

m

r

)

=

-

m

⁢

(

x

-

ξ

)

r

3

,

∂

∂

⁡

y

⁢

(

m

r

)

=

-

m

⁢

(

y

-

η

)

r

3

,

∂

∂

⁡

z

⁢

(

m

r

)

=

-

m

⁢

(

z

-

ζ

)

r

3

are the components of the gravitational with which the material point

(

ξ

,

η

,

ζ

)

acts on one mass unit in the point

(

x

,

y

,

z

)

(provided that the are chosen suitably).

The potential of a set of points

(

ξ

,

η

,

ζ

)

is the sum of the potentials of individual points, i.e. it may lead to an

integral

.

We determine the potential of all points

(

ξ

,

η

,

ζ

)

of a hollow ball, where the matter is located between two

concentric

spheres with radii

R

0

and

R

(

>

R

0

)

. Here the of mass is assumed to be presented by a

continuous function

ϱ

=

ϱ

⁢

(

r

)

at the distance

r

from the centre

O

. Let

a

be the distance from

O

of the point

A

, where the potential is to be determined. We chose

O

the origin and the ray

O

⁢

A

the

positive

z

-axis.

For obtaining the potential in

A

we must integrate over the ball shell where

R

0

≤

r

≤

R

. We use the

spherical coordinates

r

,

φ

and

ψ

which are tied to the

Cartesian coordinates

via

x

=

r

⁢

cos

⁡

φ

⁢

cos

⁡

ψ

,

y

=

r

⁢

cos

⁡

φ

⁢

sin

⁡

ψ

,

z

=

r

⁢

sin

⁡

φ

;

for attaining all points we set

R

0

≤

r

≤

R

,

-

π

2

≤

φ

≤

π

2

,

0

≤

ψ

<

2

⁢

π

.

The

cosines law

implies that

P

⁢

A

=

r

2

-

2

⁢

a

⁢

r

⁢

sin

⁡

φ

+

a

2

. Thus the potential is the triple integral

V

⁢

(

a

)

=

∫

R

0

R

∫

-

π

2

π

2

∫

0

2

⁢

π

ϱ

⁢

(

r

)

⁢

r

2

⁢

cos

⁡

φ

r

2

-

2

⁢

a

⁢

r

⁢

sin

⁡

φ

+

a

2

⁢

𝑑

r

⁢

𝑑

φ

⁢

𝑑

ψ

=

2

⁢

π

⁢

∫

R

0

R

ϱ

⁢

(

r

)

⁢

r

⁢

𝑑

r

⁢

∫

-

π

2

π

2

r

⁢

cos

⁡

φ

⁢

d

⁢

φ

r

2

-

2

⁢

a

⁢

r

⁢

sin

⁡

φ

+

a

2

,

(1)

where the factor

r

2

⁢

cos

⁡

φ

is the

coefficient

for the

coordinate

changing

|

∂

⁡

(

x

,

y

,

z

)

∂

⁡

(

r

,

φ

,

ψ

)

|

=

mod

|

cos

⁡

φ

⁢

cos

⁡

ψ

cos

⁡

φ

⁢

sin

⁡

ψ

sin

⁡

φ

-

r

⁢

sin

⁡

φ

⁢

cos

⁡

ψ

-

r

⁢

sin

⁡

φ

⁢

sin

⁡

ψ

r

⁢

cos

⁡

φ

-

r

⁢

cos

⁡

φ

⁢

sin

⁡

ψ

r

⁢

cos

⁡

φ

⁢

cos

⁡

ψ

0

|

.

We get from the latter integral

∫

-

π

2

π

2

r

⁢

cos

⁡

φ

⁢

d

⁢

φ

r

2

-

2

⁢

a

⁢

r

⁢

sin

⁡

φ

+

a

2

=

-

1

a

⁢

/

φ

=

-

π

2

π

2

⁡

r

2

-

2

⁢

a

⁢

r

⁢

sin

⁡

φ

+

a

2

=

1

a

⁢

[

(

r

+

a

)

-

|

r

-

a

|

]

.

(2)

Accordingly we have the two cases:

1

∘

.  The point

A

is outwards the hollow ball, i.e.

a

>

R

.  Then we have

|

r

-

a

|

=

a

-

r

for all

r

∈

[

R

0

,

R

]

.  The value of the integral (2) is

2

⁢

r

a

, and (1) gets the form

V

⁢

(

a

)

=

4

⁢

π

a

⁢

∫

R

0

R

ϱ

⁢

(

r

)

⁢

r

2

⁢

𝑑

r

=

M

a

,

where

M

is the mass of the hollow ball. Thus

the potential outwards the hollow ball is exactly the same as in the case that all mass were

concentrated

to the centre

. A correspondent statement concerns the attractive

V

′

⁢

(

a

)

=

-

M

a

2

.

2

∘

.  The point

A

is in the cavity of the hollow ball, i.e.

a

<

R

0

.  Then

|

r

-

a

|

=

r

-

a

on the

interval of integration

of (2). The value of (2) is equal to 2, and (1) yields

V

⁢

(

a

)

=

4

⁢

π

⁢

∫

R

0

R

ϱ

⁢

(

r

)

⁢

r

⁢

𝑑

r

,

which is on

a

. That is,

the potential of the hollow ball, when the of mass depends only on the distance from the centre, has in the cavity a constant value, and the hollow ball influences in no way on a mass inside it

.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset II

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Title

potential of hollow ball

Canonical name

PotentialOfHollowBall

Date of creation

2013-03-22 17:16:46

Last modified on

2013-03-22 17:16:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 28A25

Classification

msc 26B10

Classification

msc 26B15

Related topic

JacobiDeterminant

Related topic

ChangeOfVariablesInIntegralOnMathbbRn

Related topic

SubstitutionNotation

Related topic

ModulusOfComplexNumber

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