Page 522 - Physics Coursebook 2015 (A level)

Cambridge International A Level Physics
Intensity is the power per unit cross-sectional area.
WORKED EXAMPLE
510
body
flesh
bone
Decreasing intensity
You should recall from Chapter 13 that the intensity of a beam of radiation indicates the rate at which energy is transferred across unit cross-sectional area. Intensity is defined thus:
We can determine the intensity I using the equation: P
The attenuation (absorption) coefficient of bone is 600 m−1 for X-rays of energy 20 keV. A beam of such X-rays has an intensity of 20 W m−2. Calculate the intensity of the beam after passing through a 4.0 mm thickness of bone.
Step1 Writedownthequantitiesthatyouaregiven; make sure that the units are consistent.
I0 = 20Wm−2
x = 4.0mm = 0.004m
μ=600m−1
Step2 Substituteintheequationforintensityand solve.
Hint: Calculate the exponent (the value of –μx) first. I=I0e−μx
=20×e−(600×0.04) =20×e−2.4
=1.8Wm−2
So the intensity of the X-ray beam will have been reduced to about 10% of its initial value after passing through just 4.0 mm of bone.
Half thickness
If we compare the graphs (or equations) for the attenuation of X-rays as they pass through a material with the decay of a radioactive nuclide we see that they are both exponential decays. From Chapter 31, you should become familiar
with the concept of the half-life of a radioactive isotope (the time taken for half the nuclei in any sample of the isotope to decay). In a similar manner we refer to the half- thickness of an absorbing material. This is the thickness of material which will reduce the transmitted intensity
of an X-ray beam of a particular frequency to half its original value.
QUESTIONS
I=A
where P is power and A is the cross-sectional area normal to the radiation. The unit of intensity is W m−2.
The intensity of a collimated beam of X-rays (i.e. a beam with parallel sides, so that it does not spread out) decreases as it passes through matter. Picture a beam entering a block of material. Suppose that, after it has passed through 1 cm of material, its intensity has decreased to half its original value. Then, after it has passed through 2 cm, the intensity will have decreased to one quarter of its original value (half of a half), and then after 3cm it will be reduced to one eighth. You should recognise this pattern (1, 12 , 14 , 18 , ...) as a form of exponential decay.
We can write an equation to represent the attenuation of X-rays as they pass through a uniform material as follows:
I = I0 e−μx
where I0 is the initial intensity (before absorption), x is
the thickness of the material, I is the transmitted intensity and μ is the attenuation (or absorption) coefficient of the material. Figure 32.6 shows this pattern of absorption. It also shows that bone is a better absorber of X-rays than flesh; it has a higher attenuation coefficient. (The attenuation coefficient also depends on the energy of the X-ray photons.)
The unit of the attenuation coefficient μ is m−1 (or cm−1 etc.).
Now look at Worked example 1.
air
0
3
Use the equation I = I0 e−μx to show that the half-thickness x1⁄2 is related to the attenuation coefficient μ by:
x1⁄2 = ln2 μ
An X-ray beam transfers 400 J of energy through an area of 5.0 cm2 each second. Calculate its intensity inWm−2.
An X-ray beam of initial intensity 50 W m−2 is incident on soft tissue of attenuation coefficient 1.2 cm−1. Calculate its intensity after it has passed through a 5.0 cm thickness of tissue.
1
4 05
Distance into tissue, x
Figure 32.6 The absorption of X-rays follows an exponential pattern.
Intensity of X-rays, I