Gravitation – Potential and Potential Energy

Gravitational Potential and Potential Energy

Gravitational potential at a point is the amount of work needed to bring a unit mass from infinity to that point.

Gravitational potential is defined to be negative since gravity is attractive in nature. A negative potential means that no external work is needed to bring a unit mass from infinity to that point, since the gravity-producing mass would be doing the work to pull the unit mass to that point.

Mathematically,

\begin{aligned} \phi = -\frac{GM}{r} \end{aligned}

Note that potential is a scalar quantity. Hence, the potential at a point is simply the algebraic sum of the potential of the different masses at that point.

\begin{aligned} \phi_\text{sum} = -\frac{GM_1}{r_1}  - \frac{GM_2}{r_2} - ... \end{aligned}

Similarly, gravitational potential energy is defined as

Gravitational potential energy of a mass at a point is the amount of work needed to bring the mass from infinity to that point.

\begin{aligned} \text{GPE} = -\frac{GMm}{r^2} \end{aligned}

You may find this concept similar to gravitational force and field strength. Both gravitational force and potential energy invloves the product of two masses \begin{aligned} Mm \end{aligned} while field strength and potential involves just the gravity-producing mass \begin{aligned} M \end{aligned}.

Gravitational Potential vs Field Strength

It is easy to compare the relative values of potential and field strength because of the similar form of equations.

\begin{aligned} g &= -\frac{GM}{r^2} \\ \phi &= -\frac{GM}{r}\end{aligned}

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GPE = mgh

In many junior Physics text, gravitational potential energy is quoted with the formula
\begin{aligned} \text{GPE} = mgh \end{aligned}
This formula assumes that the change in height in insignificant compare to the radius of the Earth. This formula also calculates the change in the potential energy due to a change in position.

The formula \begin{aligned} F_G = \frac{GMm}{r} \end{aligned} calculates the actual amount of potential energy a mass possess due to its position. This formula does not calculate the change in potential energy. To calculate the change in potential energy,

\begin{aligned}\text{change in GPE} &= \frac{GMm}{r_1} - \frac{GMm}{r_2}\\\end{aligned}

If \begin{aligned} r_1 \approx r_2 \end{aligned} ,

\begin{aligned} \text{change in GPE} &= GMm ( \frac{1}{r_1} - \frac{1}{r_2} ) \\&= GMm (\frac{r_2-r_1}{r_1 r_2})\\&=gm(r_2-r_1)\\&=mgh\end{aligned}

From this, we have our old formula \begin{aligned} \text{GPE} = mgh \end{aligned}

Summary

  1. Gravitational potential at a point is the amount of work needed to bring a unit mass from infinity to that point.
  2. Gravitational potential energy of a mass at a point is the amount of work needed to bring the mass from infinity to that point.
  3. \begin{aligned} \phi = -\frac{GM}{r} \end{aligned}
  4. The gravitational potential has a larger magnitude than the field strength.
  5. When the change in height is small, we may use \begin{aligned} \text{change in GPE} = mgh \end{aligned}

Gravitation – Force and Field Strength

Concept of a Gravitational Field and Force

A gravitational field is a region in which a mass experiences a force.

A mass, \begin{aligned} m \end{aligned} that is present inside a gravitational field experiences a gravitational force. This gravitational field is produced by another mass \begin{aligned} M \end{aligned} .

The amount of force experienced by the mass \begin{aligned} m \end{aligned} is directly proportional to the product of the two masses and indirectly proportional the the square of the separation.

\begin{aligned} F_G=\frac{GM m}{r^2} \end{aligned}

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The separation \begin{aligned} r \end{aligned} is the distance between the centre of mass of the two masses. We often assume that the two masses are point masses if the separation is large relative to the radius of the masses. If the separation is not large, then it is important to use the distance between the centre of mass of the two masses. We should not use the separation between the two surfaces of the masses. Furthermore, one may safely assume that for a uniform sphere, the centre of mass is the centre of the sphere.

There are two regions about the field produced by a mass: the region outside the mass and the region inside. The region outside the mass follows the inverse exponential relationship of 1/r^2.

Inside the mass, the relation is direct proportional to the distance from the centre of the mass. This is because as you proceed nearer to the centre of the mass, there is less mass “below” you. The part of the mass “above” you pulling you “up” is offset by the mass “below” you pulling you down.

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Gravitational Field Strength

Gravitational field strength is often misunderstood. Its definition is

Gravitational field strength at a point is the gravitational force acting on a unit mass at that point.

From the definition,

\begin{aligned} \text{force} &= m \times g\\mg &= \frac{GMm}{r^2}\\g &= \frac{GM}{r^2}\end{aligned}

We can observe that gravitational field strength only depends on the gravity-producing mass and the distance from it. It is not dependent on the test mass.

There is only one value of gravitational field strength at any particular point since we are always comparing the gravitational force on one unit mass. If there are multiple masses creating gravitational fields, the gravitational field strength at any particular point would be the vector sum of all the field strengths due to the the different masses.

Field Strength on the surface of the Earth

When we calculate weight of an object, we always use the formula \begin{aligned} w = mg \end{aligned}

\begin{aligned} g \end{aligned} is referred to as the gravitational field strength (although it is commonly stated as the gravitational acceleration). Near the Earth’s surface, the field strength of the Earth is

\begin{aligned} g &=G\frac{M}{r_\text{Earth radius}^2}\\&=G\frac{M}{6400000^2}\end{aligned}

The field strength at 10 km above the surface would be \begin{aligned} G\frac{M}{6410000^2} \end{aligned}.

The difference among them is negligible. Hence we assume that the gravitational field strength on Earth’s surface is constant.

Activity 1

Access the online PHET simulation.

Objective:
What is the relationship between the force acting on m2 by m1 and the force of m1 on m2?

Task:

  1. Using the default values, observe the force on m2 by m1 and the force on m1 by m2.
    What do you observed?
  2. Change the mass of m1 to another value.
    What do you observed?
  3. Explain.

Activity 2

Access the same simulation as activity 1. Also open this activity sheet to download a copy of the activity on iCloud.

Objective:
What is the relationship between the field strength with distance?

Task:

  1. Change m2 to 1 kg, while m1 remains as 50 kg.
  2. Drag m2 to the 10 m mark, and m1 to the 0 m mark. You may not be able to do this exactly, but an approximate position would be fine.
  3. Record the force on m2 by m1 in the table.
  4. Repeat this for distances 9, 8, 7, 6, 5, 4, 3, 2 and 1 m.
  5. Observe the shape of the graph.

Question:

  1. Will the gravitational field strength ever reaches zero?
  2. At which point(from 0 m to infinity) is the field strength the strongest?

Summary

  1. understand the concept of a gravitational field as an example of a field of force and define gravitational field strength as force per unit mass.
  2. understand that, for a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre.
  3. recall and use Newton’s law of gravitation in the form \begin{aligned} F=G\frac{Mm}{r^2} \end{aligned}

Review

Question 1

Geostationary satellites are satellites that orbit around the Earth with a period of 24 hours. The satellite would appear as a stationary point relative to an observer on Earth. Calculate the distance above the Earth’s surface of a geostationary satellite. Properties of Earth

Solution

Since the centripetal force of the satellite is due to gravity,

\begin{aligned} \frac{GMm}{r^2} &= mr\omega ^2\\\frac{GM}{r^2} &= r \omega ^2 \\r &= \sqrt[3]{\frac{GM}{w}} \end{aligned}

\omega is known, since the geostationary satellite must make one orbit in one day,

\begin{aligned} \omega &= \frac{2 \pi}{24 \times 3600}\\&= 7.3\times 10^{-5} \text{ rad s}^{-1}\\r&=42.1\times 10^6\text{ m}\end{aligned}

Hence, the distance above Earth’s surface is

\begin{aligned} d &= (42.1 - 6.38) \times 10^6 \text{ m}\\ &= 35.7 \times 10^3 \text{ km} \end{aligned}

It is interesting to note that the distance of Moon from Earth is about 370 \times 10^3 \text{ km}. Hence, a geostationary satellite is about 10% of the distance from Earth to the Moon.

Work done in a Capacitor

Capacitor is an electrical component that stores charge. As charges are stored, potential energy in the capacitor also increases. In this post, I would like to explain how to calculate the energy stored in a capacitor.

Question

The switch is closed at time t = 0 s. During the time interval from t = 0.0 s to t = 4.0 s, 15 mC charge passes through the resistor.

Calculate the energy transferred by the battery during this 4.0 seconds and the energy stored in the capacitor after 4.0 s.

Capacitor Circuit

Definition of Potential Difference

In many textbooks, we learnt that the potential difference across a component is the energy released for every unit charge passing through it.

V=\frac{W}{Q}

If we are to calculate the energy released by the battery, it would be

W=V \times Q = 9.0 \times 15 \times 10^{-3} =0.135 \text{J}

Energy stored in the capacitor

Image 2

However, the energy stored in the capacitor cannot be 0.135 J. This is because a capacitor’s potential difference changes as it stores charge. The equation above requires that the p.d. remains constant. Hence, we need to use another formula.

Since capacitance is defined as the charge stored for every unit potential difference,

Q=VC

i.e. the charge stored is directly proportional to the potential difference across it. Since the work done is the area under a charge-voltage graph, we will use the formula for a triangle.

W=\frac{1}{2}QV

At time t = 4.0 s,

V=\frac{15 \times 10^{-3}}{2000 \times 10^{-6}}=7.5 \text{V}

Hence,

W=\frac{1}{2} \times 15 \times 10^{-3} \times 7.5 = 0.05625 \text{J}

Lost energy?

The energy stored in the capacitor(0.05625 J) is not the same as the energy transferred from the battery(0.135 J) because heat is lost through the resistor as current flows.

 

Radioactive Decay is Random

Radioactive decay is both random over space and time, and radioactive decay is spontaneous.

Random

Random over space means that given a large number of radioactive nuclei, it would be impossible to predict which nuclei would decay next. Random over time means that you cannot predict when a nuclei would decay at any time.

Given a large number of nuclei, it can be safe to predict that approximately half of the original radioactive nuclei would decay after one half-life. However, the exact number of nuclei decay would not be identical given two samples of equal number of original radioactive nuclei.

Continue reading “Radioactive Decay is Random”

Ultrasound Calculations

The table provides the data for the acoustic impedance and absorption coefficients for muscle and bone.

ultrasound 4

A parallel beam of ultrasound of intensity I_0  enters the muscle of thickness 3.0 cm as shown. The ultrasound is then reflected at the tissue boundary and returns to the surface of the muscle.

Calculate the intensity, in terms of I_0 , that is received when the ultrasound returns back to the surface of the muscle.


There are three important parts to solve this problem. The first part would be the attenuation of intensity inside the muscle. The second part would be the reflection of ultrasound at the muscle-bone boundary. The last part would be the intensity attenuation inside the muscle back to the surface.

I=I_0 e^{-21 times 0.03}

 

Cambridge A-Level Definitions

This post contains the definitions used in the A-level curriculum.

A-level Definitions

Motion in a circle

Radian

One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius.

Angular speed

The angular speed is defined as the angle swept out by the radius per second.

Angular velocity

The angular velocity is defined as the angle swept out in a certain direction (e.g. clockwise or anticlockwise) by the radius per second.

Simple Harmonic Motion

Simple harmonic motion

Simple harmonic motion is defined as the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point and is directed towards the point.

Period

Period is the time taken for the oscillator to complete one oscillation.

Frequency

Frequency is the number of oscillations completed per unit time

Displacement

Displacement is the distance of the oscillator from equilibrium position

Amplitude

Amplitude is the maximum displacement of the oscillator from equilibrium position

Free oscillation

An object is said to undergo free oscillation if the only external force acting on it is the restoring force

Damped oscillation

An object is said to undergo damped oscillation if friction and other forces, other than the restoring force, also acts on the object such that the oscillator energy is eventually converted to heat.

Resonance

Resonance occurs when the natural frequency of vibration of an object is equal to the driving frequency, giving a maximum amplitude of vibration

Phase

Phase refers to the point that an oscillating mass has reached within the cycle of a complete oscillation

Phase difference

Phase difference is the fraction that one wave is out of step with another wave. It can be measured in fraction, radians or degrees

Gravitational Field

Newton’s law of gravitation

Newton’s law of gravitation states that two point masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of their separation

Gravitational field

Gravitational field is a region which a mass experiences a gravitational force

Gravitational field strength

The gravitational field strength at a point is defined as the force per unit mass acting on a small mass placed at that point

Gravitational potential

Gravitational potential at a point is defined as the work done by an external agent in bringing unit mass from infinity to the point

Kepler’s third law of planetary motion

For planets describing circular orbits about the same central body, the square of the period is proportional to the cube of the radius of the orbit

Geostationary orbit

A geostationary orbit is an equatorial orbit with the same period of revolution as the Earth rotation and move in the same direction as the Earth rotation

Electric field

Law of electrostatic charges

Like charges repel, unlike charges attract

Coulomb’s law

The force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them

Electric field

Electric field is a region which a charge experiences an electric force

Electric field strength

Electric field strength at a point is the force per unit charge acting on a small positive charge placed at that point

Electric potential

Electric potential at a point is defined as the work done by an external agent in bringing unit positive charge from infinity to that point

Relationship between field strength and potential

The field strength is equal to the negative of the potential gradient at that point

Capacitance

Capacitance is the ratio of charge to potential for a conductor

Farad

One farad is one coulomb per volt

Factors affecting capacitance of a pair of parallel plates

The capacitance is directly proportional to the area of the plates and inversely proportional to the distance between them

Relative permittivity

The relative permittivity is defined as the capacitance of a parallel-plate capacitor with a dielectric between the plates divided by the capacitance of the same capacitor with a vacuum between them,

Time constant of a capacity

The time constant of a capacitor is the time for the charge to decrease to 1/e of its initial charge

Temperature

Kelvin

One kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water

Equation of state of a gas

The volume of a gas is inversely proportional to its pressure, provided that the temperature is held constant

Charles’ law

For a given mass of gas maintained at constant pressure, the volume of the gas is directly proportional to its thermodynamic temperature

Pressure law/Gay-Lussac’s law

For a given mass of gas maintained at constant volume, the pressure of the gas is directly proportional to its thermodynamic temperature

Mole

The mole is the amount of the substance which contains as many elementary particles as there are atoms in 12 g of carbon-12

Avogadro constant

Avogadro constant is the number of atoms in 12 g of carbon-12. It has a value of 6.02 × 1023 per mole.

Ideal gas

An ideal gas is a gas which obeys the equation of state pV = nRT at all temperatures, pressures and volumes

Root-Mean-Square speed

R.M.S. speed of a molecule is the average speed of an ensemble of molecules obtained by the square root of the mean of the squared-speeds of all the molecules

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Mean of the square speeds of all the molecules in the gas system

Assumptions of an ideal gas

  1. All molecules behave as identical, hard, perfectly elastic spheres.

  2. The volume of the molecules is negligible compared with the volume of the container.

  3. There are no forces if attraction or repulsion between molecules.

  4. There are many molecules, all moving randomly.

First law of thermodynamics

The increase in internal energy of a system is equal to the sum of heat added to the system and the work done on it

Alternating Current

Root-Mean-Square value

The r.m.s. value of the current or voltage is that value of the direct current or voltage that would produce heat at the same rate in a resistor

Rectification

Rectification is the process of converting an alternating current to a direct current

Smoothing

Smoothing is the process of reducing the fluctuations in the unidirectional output of an a.c. voltage

Magnetic field

Law of magnets

Like poles repel. Unlike poles attract

Motor effect

Motor effect is the phenomenon when a current-carrying conductor is at an angle to a magnetic field and experiences an electromagnetic force

Magnetic field

Magnetic field is a region which a magnetic pole experiences a magnetic force

Magnetic flux

Magnetic flux is the product of the magnetic flux density and the area normal to the lines of flux

Weber

One weber is the magnetic flux such that when linking to a circuit of one turn, an e.m.f. of 1 V will be induced in the circuit when the magnetic flux is reduced at a uniform rate to zero in one second

Magnetic flux density

Magnetic flux density is numerically equal to the force per unit length acting on a conductor carrying unit current at right angle to a magnetic field

Tesla

One tesla is the magnetic flux density which, acting normally to a long straight wire carrying 1 A current, causes a force per unit length of 1 N m–1 on the conductor

Magnetic flux linkage

Magnetic flux linkage is the product of the number of turns of a conductor and the magnetic flux passing through the conductor

Faraday’s law of electromagnetic induction

The e.m.f. induced is proportional to the rate of change of magnetic flux linkage

Lenz’s law

The direction of the induced e.m.f. is such as to cause effects to oppose the change producing it

Photoelectric effect

Photoelectric effect

Photoelectric effect is the emission of electrons from the surface of a metal when electromagnetic radiation is incident on its surface

Observations from photoelectric effect

  • If photoemission takes place, it does so instantaneously. There is no delay between illumination and emission.

  • Photoelectric emission takes place only if the frequency of the incident radiation is above a certain minimum value called the threshold frequency.

  • Different metals have different threshold frequencies.

  • Whether or not emission takes place depends only on whether the frequency of the radiation is above the threshold for that surface. It does not depends on the intensity of radiation.

  • For a given frequency, the rate of emission of photoelectrons is proportional to the intensity of the radiation.

Conclusion from photoelectric effect

  • The photoelectrons have a range of kinetic energies, from zero up to some maximum value. If the frequency of the incident radiation is increased the maximum kinetic energy of the photoelectrons also increases.

  • For constant frequency of the incident radiation, the maximum kinetic energy is unaffected by the intensity of the radiation.

  • When the maximum kinetic energy of the photoelectrons is zero, the minimum frequency required to cause emission from the surface may be found.

Photon

A photon refers to a quantum of energy when the energy is in the form of electromagnetic radiation

Work function

Work function energy is the minimum amount of energy necessary for an electron to escape from the surface of a metal

Nuclear Physics

Half-life

The half-life of a radioactive nuclide is the time taken for the number of undecayed nuclei to be reduced to half its original number

Activity

The activity of a radioactive source is the number of nuclear decays produced per unit time in the source. Activity is measured in becquerels (Bq), where 1 Bq is 1 decay per second

Decay constant

Decay constant is defined as the probability per unit time that a nucleus will undergo decay

Atomic mass unit

One atomic mass unit is defined as being equal to 1/12 of the mass of a carbon-12 atom

Mass defect

Mass defect of a nucleus is the difference between the total mass of the separate nucleons and the combined mass of the nucleus

Binding energy

Binding energy is the energy equivalent of the mass defect of a nucleus. It is the energy required to separate to infinity all the nucleons of a nucleus

Binding energy per nucleon

Binding energy per nucleon is defined as the total energy needed to completely separately all the nucleons in a nucleus divided by the number of nucleons in the nucleus

Nuclear fission

Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei of approximately the same mass

Observation from probing matter with alpha particles(Rutherford scattering)

  • the vast majority of the alpha-particles passed through the foil with very little or no deviation from their original path

  • a small number of particles were deviated through an angle of more than about 10º.

  • an extremely small number of particles (one in ten thousand) were deflected through an angle greater than 90º.

Conclusions from probing matter with alpha particles

  • the majority of the mass of an atom is concentrated in a very small volume at the centre of the atom. Most alpha-particles would therefore pass through the foil undeviated.

  • the centre of an atom is positively charged. Alpha-particles which are also positively charged, passing close to the nucleus will experience a repulsive force causing them to deviate.

  • Only alpha-particles that pass very close to the nucleus, almost striking it head-on, will experience large enough repulsive forces to cause them to deviate through angles greater than 90º. The fact that so few particles did so confirms that the nucleus is very small, and that most of the atom is empty space.

  • Because atoms are neutral, the atoms must contain negative particles. These travel around the nucleus

Direct Sensing

Properties of an ideal op-amp

  • infinite input impedance

  • zero output impedance

  • infinite open-loop gain

  • infinite bandwidth

  • infinite slew rate

Virtual earth

In a inverting negative feedback circuit, the negative input voltage would tend towards the value of the non-inverting input, which is earthed. Hence, the inverting input is very close to 0 V. This input is said to be virtual earth.

Remote Sensing

Intensity

Intensity is the wave power per unit area normal to wave direction

Hardness(X-ray)

Hardness is the penetration of the X-ray beam, which determines the fraction of the intensity of the incident beam that can penetrate the part of the body being X-rayed

Sharpness(X-ray)

Sharpness is related to the ease with which the edges of structure can be determined

Contrast(X-ray)

Contrast refers to the range of blackening in an X-ray image. Good contrast has a wide range of blackening

Half-value thickness

Half-value thickness is the thickness of medium that will reduce the intensity of X-ray beam by half

Tomography

Tomography is a technique whereby a three-dimensional image through a body may be obtained

Specific acoustic impedance

Specific acoustic impedance is the product of the density of the medium and the speed of the wave in the medium

Communication

Modulation

Modulation is the variation of either the amplitude or frequency of the carrier wave

Amplitude modulation

In amplitude modulation, the carrier wave has constant frequency. The amplitude of the carrier wave is made to vary. These variations are in synchrony with the displacement of the information signal

Frequency

In frequency modulation, the amplitude of the carrier wave remains constant. The frequency of the carrier wave is made to vary in synchrony with the displacement of the information signal

Attenuation

Attenuation is the decrease in signal strength as it travels over a distance

Bandwidth

The bandwidth is the range of frequencies that the signal occupies

Analogue signal

An analogue signal can have any value, within limits, and is an exact representation of the raw information

Digital signal

A digital signal consists of a series of ones and zeros, with no values between them

Bit

Each digit in the binary number is known as a bit

Noise

Noise is the random, unwanted signal that adds to and distorts a transmitted signal

Sampling

Sampling is the measurement of the analogue signal at regular time intervals

Analogue-to-digital converter

In an ADC, the analogue voltage is sampled at regular intervals of time, at what is known as the sampling frequency or sampling rate. The value of the sample voltage measured at each sampling time is converted into a digital number that represents the voltage value

Digital-to-analogue converter

In a DAC, a digital signal is converted into an analogue signal

Characteristics of wire-pair cable

  • used mainly for short-distance communication

  • cause high attenuation of signal

  • easily pick up noise

  • suffer from cross-talk and are of low security

  • have limited bandwidth

 

Comparing a wire pair and coaxial cable

The coaxial cable

  • is more costly

  • causes less attenuation

  • is less noisy and is more secure

  • has a larger bandwidth

Advantages of fibre optics

  • large bandwidth, giving rise to large transmission capacity

  • much lower cost than metal wire

  • diameter and weight of cable is much less than metal cable, hence easier handling and storage

  • much less signal attenuation, so far fewer regenerator amplifiers are required, reducing the cost of installation

  • do not pick up electromagnetic interference, so very high security and negligible cross-talk

  • can be laid alongside existing routes such as electric railways lines and power

Geostationary satellites

Geostationary satellites are satellites orbiting the Earth with a period of 24 hours at a height of 3.6 × 104 km above the Earth’s surface. The satellites orbit in the same direction as the rotation of the Earth (from west to east) and the orbit is above the equator

Polar satellites

Polar satellites are satellites that have low orbits and pass over the poles

Types of Ultrasound Scans

We will investigate two types of ultrasound scans.

A-Scan

This is a simple type of scan. A pulse of ultrasound is sent into the tissue and the echoes are detected to determine the depth of the structure.

ultrasound 3

The thickness of the bone can be calculated using Δt , which is the time it takes for the ultrasound to travel twice the distance of the bone. Since the speed of ultrasound in bone is known, the thickness of the bone can be determined using the formula

t=frac{v times Delta t}{2}

where v is the speed of ultrasound in the medium.

A-scans are used for simple procedures such as measuring the thickness of bones.

B-Scan

This type of scan produces more detailed images than A-scan. A series of pulses are directed at the organ, and each reflected pulse is analysed to the depth and nature of the reflecting surface. The timing of the reflected pulse provides information to the depth of the surface while the intensity provides information to the type of reflecting surface.

ultrasound 4

In the above diagram, it can be seen that when the series of pulses are sent, a pattern of dots is obtained, which allows the computer to generate an image of the scanned organ.

Ultrasound Intensity Reflection and Attenuation

Intensity Reflection Coefficient

When a beam of ultrasound is directed into a body, a certain proportion of the initial intensity would be reflected at the boundary. The ratio of the reflected intensity to the initial intensity can be obtained using the formula

\begin{aligned}\frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}\end{aligned}

ultrasound

Air-Tissue Boundary

Using the formula from above, we can calculate that the intensity reflection coefficient of an air-tissue boundary is more than 99%.

\begin{aligned}\frac{I_r}{I_0}=\frac{(1.63 - 0.0004)^2}{(1.63 + 0.0004)^2}=0.999\end{aligned}

Note that the squaring of both the numerator and the denominator means that it does not matter which medium is assigned as Z_1 and Z_2 .

This is the reason why impedance matching need to be done when ultrasound is performed on a patient.

Intensity Attenuation

When ultrasound travels across a tissue, its intensity attenuates. This is because the wave loses energy to the tissue as it travels. The intensity of the wave at any particular depth of the tissue is determined by the formula

I=I_0 e^{-mu x}

where x is the depth of the point at which the intensity is determined, $latex μ $ is the absorption coefficient of the material and I_0 is the original intensity before the ultrasound wave enters the particular medium.

ultrasound 2

To find the intensity of the ultrasound that is received at the edge of medium 1, we can employ the following sequence of calculations:

  1. I_0 is the initial intensity.
  2. When the wave reaches the boundary of medium 1 and medium 2, it would have been attenuated by the amount I=I_0 e^{-\mu x} where x  is the depth of medium 1.
  3. This wave is then reflected at the boundary, where \begin{aligned} I_2 = \frac{(Z_2-Z_1)^2}{Z_2+Z_1)^2}\end{aligned}.
  4. Lastly, when the wave reaches the edge of medium 1, its final intensity I_3=I_2 e^{-\mu x}

Summary

  1. When you are calculating the reflection of ultrasound at a boundary, use \frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}
  2. When you are calculating the intensity attenuation over a distance, use I=I_0 e^{-\mu x}

Problems

Using Ultrasound for Medical Imaging

In medical imaging, ultrasound is used mainly to detect the shape of a foetus. Ultrasound is also used for imaging the internal of a body. Since the wavelength of an ultrasound is in the order of about 1 cm, it cannot be used to detect structures that are small. Ultrasound is most effective with structures like the bone, or internal organs and foetus.

Principle of Using Ultrasound for Medical Imaging

The principle used in medical imaging is called echo sounding. Ultrasound is directed into the body. Since different materials have different acoustic impedance, when an ultrasound is incident on a boundary of two materials, there would be a reflected component of the ultrasound. This reflected component would then be received by the detector and used to compose the image of the reflected surfaces in the body.

Acoustic Impedance

Acoustic impedance is defined as the product of the density of medium and the speed of sound in the medium.

Mathematically,

 Z= \rho \times c

The unit of Z would be \text{kg m}^{-2}\text{ s}^{-1}

The meaning of acoustic impedance is how good the sound wave travels in that medium. Large value means the sound wave travels better. The acoustic impedance of some common materials are listed below:

Acoustic impedance of some common materials in the body
Acoustic impedance of some common materials in the body

There will be significant reflection of the ultrasound waves at boundaries of two materials that have widely different acoustic impedance. For example, significant reflection occurs between blood and done boundaries. Hence, we can use ultrasound to investigate the shape of the bone. This would not be a good way to investigate the structure of muscle because muscle and blood have similar acoustic impedance.

Use of Gel in Ultrasonic Scanning

When a pregnant woman performs an ultrasound scan, the doctor would always smear a layer of gel onto her abdomen before performing the scan. The boundary between air and the skin causes significant reflection. The acoustic impedance of air is 0.0004 but skin has an acoustic impedance  of about 1.6. Most of the incident ultrasonic waves would be reflected, leaving very little waves to penetrate the body to be reflected by the other internal structures of the body. Using gel removes the air medium such that the boundary would be between the gel and the skin, which have very similar acoustic impedance. Smearing of gel onto the skin is known as impedance matching.