84
1 Mathematical Physics
N = 6;
y n = 29.5;
x n = 3;
x
2
n = 19
x
3
n = 27;
x
4
n = 115;
x n y n = 21.3;
x
2
n y n = 158.1
Solving (1), (2) and (3) we find a 0 = 0.582; a 1 = −1.182; a 2 = 1.556
Fig. 1.18 Least square fit of the parabola
1.102 C =
2πε 0
ln
b
a
(1)
From propagation of errors
σ c =
∂c
∂b
2
σ
2
b +
∂c
∂a
2
σ
2
a
1/2
(2)
∂c
∂b
= −
c
b ln
b
a
;
∂c
∂a
=
c
a ln
b
a
(3)
Using (1), (2) and (3) and simplifying
σ c
c
=
ln
b
a
−1/2 σ
2
a
a 2 +
σ
2
a
b 2
1/2
Substituting a = 10 mm, b = 20 mm, σ a = 1 mm and σ b = 1 mm, σ c /c =
0.16