Binomial Formula
Very often in physics we have to evaluate quantities of the form
(1)
(a +b)p
where b is small compared to a (this is written as b << a and in practice means that b is
a
smaller than
). The quantity in (1) is then very close to ap, but we often need a better
10
approximation. Factor out a to get (a +b)p = ap (1 + )p where  = b/a <<1. Then perform a
Taylor series expansion about  = 0.
2  d2
p
p
p
d
1     1    1       2 1     
 d
 0 2!  d 
  0
Evaluating the derivatives gives
1   
p
 1  p  p  p  1
2

2!
Equation (2) is called the binomial formula and the first few terms are a good
approximation whenever  is small compared to 1. Our original expression has the
approximate value:
2


b p  p  1  b 
p
(3)
 a  b   a p 1  p 
   
a
2!  a 


(2)
As an example, here’s a situation that arises all the time with curved lenses and mirrors.
Consider a small piece of a sphere. The sphere could be a lens or mirror surface. The
piece has diameter d and the sphere has radius R where d << R. Draw a perpendicular
line segment from the edge of the spherical piece to the axis of symmetry and call the
length of this segment y. The point of intersection divides the radius into two pieces, one
of length x and the other of length R-x. Estimate the fraction x/(R-x), i.e., estimate the
percentage of the short length relative to the long one.
Solution:
 R  x
2
 R2  y 2
1/ 2
 

y2
y2 
y2  
x  R  R  y  R  R 1  2  R 1  1  2   R 1  1  2  
R
R 
  R  

Now y is smaller than d, so y/R <<1 and we can apply the binomial formula to the
2
2
1/ 2

y2 
expression 1  2  to get
 R 
 
 y2 
y 2 
x
y2
x  R 1  1  2    R  2   
R 2R2
 2R 
  2R 
This says that if d is small compared to R, x is MUCH smaller compared to R.
For example, a typical glass lens might have R = 10cm, d = 1cm, so d/R = 0.1 Then x/R
would be smaller than 0.002. In practice this means that x can be regarded as effectively
zero in comparison with R.