Ch 1 · 24 Medical Physics · A Level Physics 9702
Topic 1 of 1 in Medical Physics — A Level Physics 9702 — 78 lessons.
Introduction
Hospitals use physics to see inside the body without surgery: sound waves for scans of unborn babies, X-rays and CT for bones and tumours, and antimatter for PET scans of brain activity. (Syllabus topic 24: sections 24.1–24.3.)
Key ideas — 24.1 (a) The piezo-electric transducer
A piezo-electric crystal (e.g. quartz) changes shape when a p.d. is applied across it, and generates an e.m.f. when its shape changes (it is squashed or stretched).
🧑🏫 Teacher's explanation
Imagine a sponge that reacts to your touch. If you squeeze it, it changes shape; if you tap it, it vibrates. A piezo-electric crystal is like a 'smart' version of that sponge. When you apply a voltage (p.d.) across it, the crystal physically expands or contracts. If you apply an alternating voltage (AC), the crystal vibrates back and forth rapidly, acting like a tiny loudspeaker that creates ultrasound waves.
📖 Definitions
🔍 Worked example — Easy
A piezo-electric crystal is used to generate ultrasound. If an alternating p.d. is applied to the crystal, what is the primary mechanism that produces the sound wave?
🔍 Worked example — Standard
An ultrasound pulse travels from tissue 1 (Z_1 = 1.5 × 10^6 kg m^-2 s^-1) to tissue 2 (Z_2 = 1.7 × 10^6 kg m^-2 s^-1). Calculate the intensity reflection coefficient I_R/I_0.
🔍 Worked example — Exam-style
An ultrasound beam with initial intensity I_0 = 10.0 W m^-2 travels through 5.0 cm of tissue. The attenuation coefficient μ is 0.5 m^-1. Calculate the intensity I at the end of the tissue.
⚠️ Common errors
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🎯 Exam question
(a) State the effect that allows the crystal to convert electrical energy into ultrasound waves. [1] (b) The ultrasound pulse encounters a boundary between two tissues. The acoustic impedance of tissue 1 is 1.5 × 10^6 kg m^-2 s^-1 and tissue 2 is 1.6 × 10^6 kg m^-2 s^-1. Calculate the fraction of the intensity that is reflected at this boundary. [2] [3 marks]
Key ideas — 24.1 (b) Ultrasound imaging
Pulses of ultrasound sent into the body are partly reflected at boundaries between tissues of different acoustic impedance. The time delay of each echo gives the depth of the boundary (distance = ct/2); the intensity of the echo indicates the type of boundary. Scanning the beam across builds up an image (B-scan). A coupling gel is used between the transducer and the skin, to remove air: air–skin boundaries reflect almost all the ultrasound.
🧑🏫 Teacher's explanation
Imagine you are standing in a large, dark canyon and you shout. If you hear an echo, you can estimate how far away the canyon wall is by timing how long it takes for the sound to bounce back. Ultrasound imaging works exactly like this, but instead of sound waves traveling through air, we use high-frequency waves traveling through human tissue.
📖 Definitions
🔍 Worked example — Easy
An ultrasound pulse is sent into tissue. The echo is received after a time delay of 0.40 ms. Calculate the depth of the tissue boundary if the speed of sound in the tissue is 1540 m s^-1.
🔍 Worked example — Standard
A boundary exists between tissue 1 (Z_1 = 1.5 × 10^6 kg m^-2 s^-1) and tissue 2 (Z_2 = 1.7 × 10^6 kg m^-2 s^-1). Calculate the intensity reflection coefficient I_R/I_0.
🔍 Worked example — Exam-style
An ultrasound beam of initial intensity I_0 = 100 W m^-2 travels through 5.0 cm of fat. The attenuation coefficient μ is 0.5 m^-1. Calculate the intensity I at the end of the 5.0 cm path.
⚠️ Common errors
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🎯 Exam question
(a) Calculate the depth of the boundary if the speed of sound in the tissue is 1550 m s^-1. [2] (b) The intensity of the pulse is I_0. At the boundary, the acoustic impedance changes from Z_1 = 1.6 × 10^6 kg m^-2 s^-1 to Z_2 = 1.7 × 10^6 kg m^-2 s^-1. Calculate the fraction of intensity reflected, I_R/I_0. [2] [4 marks]
Key ideas — 24.1 (c) Acoustic impedance and reflection
The specific acoustic impedance of a medium is
🧑🏫 Teacher's explanation
Imagine you are running through a crowd. If the crowd is moving at the same speed as you, you can weave through them easily (high transmission). But if you suddenly hit a wall of people standing perfectly still and packed tightly, you'll bounce back (high reflection). In ultrasound, the 'crowd' is the medium (like tissue), and the 'wall' is a boundary where the density or speed of sound changes abruptly.
📖 Definitions
🔍 Worked example — Easy
An ultrasound wave travels through soft tissue with a density of 1010 kg m^-3 and a speed of sound of 1540 m s^-1. Calculate the specific acoustic impedance Z of the tissue.
🔍 Worked example — Standard
An ultrasound pulse hits a boundary between soft tissue (Z_1 = 1.6 × 10^6 kg m^-2 s^-1) and bone (Z_2 = 6.0 × 10^6 kg m^-2 s^-1). Calculate the fraction of intensity reflected, I_R/I_0.
🔍 Worked example — Exam-style
An ultrasound beam with initial intensity I_0 = 1.0 W m^-2 enters a medium with an attenuation coefficient μ = 0.5 m^-1. Calculate the intensity I after traveling 4.0 m through the medium.
⚠️ Common errors
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🎯 Exam question
Key ideas — 24.1 (d) Attenuation
As ultrasound (or X-rays) pass through matter, the intensity decreases exponentially with thickness:
🧑🏫 Teacher's explanation
Imagine you are walking through a dense crowd. The further you walk, the more people you bump into, and the harder it becomes to see the person behind you. In physics, as a beam of waves (like ultrasound or X-rays) travels through a material, it doesn't just lose energy linearly; it loses it exponentially. This is because every time a wave travels a certain distance, a certain percentage of the remaining waves are scattered or absorbed.
📖 Definitions
🔍 Worked example — Standard
An X-ray beam passes through a lead plate. The initial intensity is 500 Gy/s and the intensity after 2.0 cm is 50 Gy/s. Calculate the attenuation coefficient μ in cm^-1.
⚠️ Common errors
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🎯 Exam question
Key ideas — 24.2 (a) Producing X-rays
In an X-ray tube, electrons emitted by a heated cathode are accelerated through a large p.d. V (tens of kV) and bombard a metal target (e.g. tungsten). As they decelerate, X-ray photons are produced (most of the energy becomes heat, so the target is cooled, often by rotating it).
🧑🏫 Teacher's explanation
Imagine a high-speed collision between a car and a brick wall. If the car has a massive amount of kinetic energy and hits the wall, that energy doesn't just vanish; it is converted into sound, heat, and structural deformation. In an X-ray tube, we replace the car with an electron and the wall with a metal target. The electron is 'thrown' at the target using a huge voltage V. When the electron slams into the metal atoms, it slows down abruptly. This sudden loss of kinetic energy is converted into high-energy electromagnetic radiation: X-rays.
📖 Definitions
🔍 Worked example — Easy
An electron is accelerated through a potential difference of 50.0 kV. Calculate the minimum wavelength λ_min of the X-rays produced. (Use h = 6.63 × 10^-34 J s, c = 3.00 × 10^8 m s^-1, e = 1.60 × 10^-19 C)
🔍 Worked example — Standard
An X-ray beam passes through a tissue of thickness x = 5.00 cm with a linear attenuation coefficient μ = 0.250 cm^-1. If the initial intensity is I_0 = 100 W m^-2, calculate the transmitted intensity I.
🔍 Worked example — Exam-style
In a CT scan, multiple X-ray images are taken from different angles. If the total attenuation of a body part is measured to be 0.85 (meaning I/I_0 = 0.85), and the thickness is 20.0 cm, calculate the attenuation coefficient μ.
⚠️ Common errors
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🎯 Exam question
Key ideas — 24.2 (b) X-ray imaging and contrast
X-rays passing through the body are attenuated more by dense materials with high attenuation coefficients (bone) than by soft tissue. The transmitted X-rays fall on a detector, making a shadow image.
Key ideas — 24.2 (c) CT scanning
CT distinguishes soft tissues much better than a single X-ray, but gives a higher radiation dose.
🧑🏫 Teacher's explanation
Imagine you are trying to understand the shape of a loaf of bread. A single X-ray is like looking at the whole loaf from one side; you see the silhouette, but you can't tell if there is a hole in the middle or where exactly the raisins are located. A CT scan is like slicing that loaf into many thin, individual slices. You look at each slice from many different angles, reconstruct a 2D picture of that specific slice, and then stack all those 2D pictures together to create a complete 3D model of the bread.
📖 Definitions
🔍 Worked example — Easy
A CT scanner takes 180 X-ray projections of a single slice of a patient's organ from different angles. If the computer combines these to form a 2D image, explain why this is superior to a single X-ray for identifying a small tumor.
🔍 Worked example — Standard
An X-ray beam with initial intensity I_0 passes through a tissue section. If the linear attenuation coefficient of the tissue is μ = 0.25 cm^-1 and the thickness of the section is x = 4.0 cm, calculate the transmitted intensity I.
🔍 Worked example — Exam-style
A CT scan is performed on a patient. The scan produces 50 slices of a specific organ. Each slice is a 2D image. To create a 3D volume, the computer must combine these slices. If the total volume of the organ is V, and the thickness of each slice is Δ x, express the total length L of the organ in terms of V and Δ x assuming the cross-sectional area A is constant.
⚠️ Common errors
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🎯 Exam question
(a) Explain how these images are used to produce a 3D image of the brain. [2] (b) A specific tissue in the brain attenuates X-rays such that the intensity decreases by a factor of e over a distance of 5.0 cm. Calculate the linear attenuation coefficient μ for this tissue. [2] [4 marks]
Key ideas — 24.3 PET scanning
A tracer is a substance containing radioactive nuclei that can be introduced into the body (e.g. injected) and is then absorbed by the tissue being studied (for example, fluorine-18 attached to glucose, taken up by active brain cells or tumours).
🧑🏫 Teacher's explanation
Imagine you are in a dark room with a handful of glowing marbles. If you throw a marble and it hits a wall, it might bounce back or create a flash. In medical imaging, we use 'glowing' particles called tracers to see inside the body. In PET scanning, we use a special type of particle called a positron. A positron is like the 'mirror image' of an electron; it has the exact same mass but a positive charge.
📖 Definitions
🔍 Worked example — Easy
Calculate the energy of a single gamma-ray photon produced when an electron and a positron annihilate, given the rest mass of an electron m_e = 9.11 × 10^-31 kg.
🔍 Worked example — Standard
A positron annihilates with an electron. If the total energy released is 1.64 × 10^-13 J, calculate the energy of one of the resulting gamma-ray photons.
🔍 Worked example — Exam-style
In a PET scan, a positron is emitted. It travels through tissue and annihilates with an electron. State the two conservation laws that must be satisfied during this annihilation process and explain why the photons must travel in opposite directions.
⚠️ Common errors
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🎯 Exam question
Worked example
Example 1: reflection. Find the fraction of ultrasound intensity reflected at a muscle–bone boundary. Z_muscle = 1.7 × 10^6, Z_bone = 7.8 × 10^6 kg m⁻² s⁻¹.
🔍 Worked example — Easy
An ultrasound pulse travels from muscle (Z_1 = 1.7 × 10^6 kg m^-2 s^-1) to bone (Z_2 = 7.8 × 10^6 kg m^-2 s^-1). Calculate the fraction of intensity reflected at the boundary.
🔍 Worked example — Standard
An X-ray beam with initial intensity I_0 passes through 2.0 cm of bone. If the attenuation coefficient μ of bone is 300 m^-1, calculate the transmitted intensity as a fraction of the incident intensity.
🔍 Worked example — Exam-style
Calculate the energy of a single gamma-ray photon produced by the annihilation of an electron and a positron. Use m_e = 9.11 × 10^-31 kg and c = 3.00 × 10^8 m/s.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) A crystal changes shape when a p.d. is applied and generates an e.m.f. when deformed; an alternating p.d. at ultrasonic frequency makes it vibrate and emit ultrasound. (2) To remove air: the large impedance difference between air and skin would reflect almost all the ultrasound. (3) Z = rho c, density × speed of sound in the medium. (4) The difference in brightness between regions of the image; use a contrast medium with a high attenuation coefficient (barium, iodine). (5) Many X-ray images of one slice from different angles give a 2D image of the slice; repeating along an axis and combining slices gives a 3D image. (6) The electron–positron pair has almost zero momentum, so the photons must have equal and opposite momenta; each has energy m_ec^2 = 0.51 MeV.