The Chemistry Maths Book, Second Edition

15.3 Two expansions in Legendre polynomials 423

whereR

q

is the distance of point Pfrom the charge (if a chargeq′is placed at P, the

potential energy of the system of two charges isVq′ 1 = 1 qq′ 24 πε

0

R

q

). By the cosine rule

for the triangle OPq, we haveR

2

q

1 = 1 r

2

1 + 1 R

2

1 − 12 r 1 R 1 cos 1 θ, so that (15.24) can be written as

(15.25)

We consider the caseR 1 > 1 r. We write (15.25) as

and, by equation (15.22) witht 1 = 1 r 2 Randx 1 = 1 cos 1 θ, this can be expanded in Legendre

polynomials as

(15.26)

Whenr 1 > 1 R, the same expansion is valid, but with rand Rinterchanged.

The potential of a distribution of charges

We consider a system of Ncharges,

q

1

at (x

1

, y

1

, z

1

), q

2

at (x

2

, y

2

, z

2

), =,q

n

at (x

n

,y

n

,z

n

)

as illustrated in Figure 15.2 (for 2 charges).

Each charge makes its individual contribution

to the electrostatic potential at point P, and the total

potential at P is the sum of these contributions,

(15.27)

If the point P is exteriorto the distribution of

charges, so thatR 1 > 1 r

i

, for alli, the potential can be

expanded term by term as in (15.26):

(15.28)

=

















=

+

=

∑∑

1

4

1

0

0

1

1

πε

θ

l

l

i

N

ii

l

li

R

qr P

∞

(cos )

V

q

R

r

R

P

i

N

i

l

l

i

li

=

















==

∑∑

1

0

0

4 πε

θ

∞

(cos )

V

q

R

i

N

i

i

=

=

∑

1 0

4 πε

V

q

R

r

R

P

r

R

l

l

l

=











 ,<

=

∑

4

1

0

0

πε

θ

∞

(cos )

V

q

R

r

R

r

R

=−













• 
  

























−/

4

12

0

12

2

πε

cosθ









V

q

=− +RrR r

−/

4

2

0

2212

πε

(cos)θ

.

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• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • 
        
      
      
      
      
    
    
    
    
  
  
  
  

o

q

1

q

2

p

r

1

r

2

R

R

1

R

2

...

...

...

....

...

....

....

.....

..θ

1

.

...

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θ

2

Figure 15.2