Also,
I
I 0
= exp Àlx
ð
Þ = exp À4lx H
ð
Þ
or
1 À
X
100
¼ exp À4 Â 0:693
ð
Þ¼0:0625368
or X ¼ 1 À 0:0625368
ð
Þ Â 100 ¼ 93:7%
Example 3 A beam of X-ray consists of equal intensities of wavelengths 0.064 Å
and 0.098 Å. When they pass through a piece of lead, their attenuated beam
intensity is in the ratio 3:1. The mass absorption coefficients are 0.164 m
2 /kg for
harder component and 0.35 m
2 /kg for softer component, respectively. Calculate the
thickness of lead if its density is 11340 kg/m
3 .
Solution: Given: k 1 ¼ 0.064 ˚
A ;k 2 ¼ 0.098 ˚
A ;l m1 = 0.164 m
2
kg for k 1 ;l m2 = 0.35 m
2
kg
for k 2 , I k 1
ð Þ = I k 2
ð Þ;q Pb
ð Þ¼ 11340
kg
m
3 ; also according to question I 1 : I 2 ¼
3 : 1; x ¼ ?
Here, l 1 and l 2 are :
l 1 ¼ l m1 Â q ¼ 0:164 Â 11340 ¼ 1859:76m
À1
l 2 ¼ l m2 Â q ¼ 0:35 Â 11340 ¼ 3969:00m
À1
Now, making use of the exponential equation for intensity variation and taking
the ratio, we obtain
3 ¼
exp À1859.76
ð
Þ
exp À3969.00
ð
Þ
¼ exp 2109.24
ð
Þ x
or 2109.24x = ln3
or x =
ln3
2109.24
¼ 5.22 Â 10
À4 m
Example 4 A copper sheet of 1.05 mm thick can reduce the intensity of the X-ray
beam to 7.5% of its original value. Determine the value of its mass absorption
coefficient of copper if its density is 8930 kg/m
3 .
Solution: Given: x = 1.05 mm = 1.05 Â 10
À3 m, I
I0 ¼ 7.5%;q Cu
ð Þ¼ 8930
kg
m
3 ;l m¼?
7.3 Absorption of X-Rays
283