1000 Solved Problems in Modern Physics

84 1 Mathematical Physics

N=6;

∑

yn= 29 .5;

∑

xn=3;

∑

x^2 n= 19

∑

xn^3 =27;

∑

xn^4 =115;

∑

xnyn= 21 .3;

∑

x^2 nyn= 158. 1

Solving (1), (2) and (3) we finda 0 = 0 .582;a 1 =− 1 .182;a 2 = 1. 556

Fig. 1.18Least square fit of the parabola

1.102C=^2 πε^0
ln

(b

a

) (1)

From propagation of errors

σc=

[(

∂c

∂b

) 2

σb^2 +

(

∂c

∂a

) 2

σa^2

] 1 / 2

(2)

∂c

∂b

=−

c

blnba

;

∂c

∂a

=

c

alnba

(3)

Using (1), (2) and (3) and simplifying

σc

c

=

[(

ln

b

a

)]− 1 / 2 [

σa^2

a^2

+

σa^2

b^2

] 1 / 2

Substitutinga=10 mm,b=20 mm,σa=1 mm andσb=1mm,σc/c=

0. 16