Physics is a science dealing with nature. It is concerned with the study of two concepts, matter and energy, and how they interact with each other. Matter is one, which occupies space and it is made up of molecules or atoms, e.g. copper, rubber, water and air. Matter exists in the form of solid, liquid, gas, liquid crystal and plasma state. Matter can be converted from one form to another by a physical or chemical change.
Energy is the ability to do work and it has several forms. Energy can be converted from one form to another, e.g. water heater converts electrical energy into heat energy. The law of conservation of energy states that, the total energy in the universe is constant. That means energy can neither be created nor destroyed, but it can be converted from one form to another. Matter can be converted into energy and vice versa. In general, physicist studies the behavior of matter and energy under different physical conditions.
QUANTITIES AND UNITS
To study the matter, energy and their various properties, measurements of physical quantities, such as mass, length, and time are required. Any physical quantity is measured accurately in terms of a standard of its own kind, e.g. distance is measured in meter, mass in kilogram and time in second, respectively. Therefore, the unit is a quantity adopted as a standard of measurement, in terms of which similar quantities can be measured, e.g. kilogram, meter, and second.
Fundamental Units and Derived Units
The units which are independent of one another and having their own standard (base) are called fundamental units, e.g. Mass-kilogram, Length-meter, and Time-second. The units of all other physical quantities can be obtained from the fundamental units.
The units, which are not having their own standard (base) and obtained from the fundamental units are called derived units, for example;
Area | — | meter2 |
Velocity | — | meter/second, |
Density | — | kilogram/meter3 |
Over the years, many systems of units like FPS (foot, pound, second), CGS (centimeter, gram, second), MKS (meter, kilogram, second) and rationalized MKS have been used. In 1960, the new system of units, the Systems International d'units (SI Units) was introduced. The SI system is superior to all other systems, more convenient in practice, and it is used throughout the world. There are seven fundamental units and two supplementary units in the SI system as shown in Table 1.1.
Units of Length, Mass and Time
Length: The unit of length is meter. One meter is the length equal to 1,65,0763.73 wavelength of the orange-red line of krypton-86 discharge lamp, kept at 15°C and 76 cm of mercury.
Mass: The unit of mass is kilogram. One kilogram is the mass of the platinum-iridium cylinder of diameter equal to its height kept at the International Bureau of Weights and Measures near Paris.
Time: The unit of time is second. The second is defined as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between two specified energy levels of Cesium -133 atom.
Prefixes
Though the SI units are a coherent system, they are found to be either too large or low in practice, e.g. the activity of an isotope for bone scan is expressed in billions of becquerel's. Hence, prefixes are used to overcome the above difficulty, as shown in Table 1.2. These prefixes are conveniently used to describe very large or small physical quantities. In radiation, giga becquerel (GBq), kilovolt (kV), centi gray (cGy), milliampere (mA) and nanometer (nm) are commonly used.
Conventions for SI units
- When the unit is named after a scientist, it should not be written in a capital initial letter, e.g. newton, ampere. The symbol of the unit is expressed in capital letters, e.g. N for newton
- The symbol of all other units should be written with small letters, e.g. m for meter
- Only singular form of unit is to be used, e.g. 500 meters is written as 500 m. No full stops or punctuation marks should be used at the end of the symbol5
- Space is to be left between the numerical and symbol, e.g. 20 s and not as 20s
- Mathematical indices notation should be used than slash sign (/), e.g. meters per second should be written as ms−1 not m/s
- In the temperature, for unit kelvin no degree sign is used, e.g. 273 K and not as 273° K.
Radiography units
In radiography, units like kilovoltage (kVp), milliampere (mA), milliampere second (mAs), kiloelectron volt (keV) and heat units (HU) are used. Though they are not strictly SI units, but are used in practice for convenience.
One kVp is equal to 1000 volts and refers to the potential difference applied between anode and cathode of X-ray tube and the suffix refers to the peak value of the potential difference across the X-ray tube, since the potential difference is varying in nature.
One milliampere is equal to 1/1000 of amperes and refers to the number of electrons flowing per second from cathode to anode. It is directly related to the amount of X-rays that are produced, in turn the blackening on the film. The amount of X-rays produced or blackening of the film also depends on the duration of electron flow or exposure time in seconds. Hence, mAs means milliampere seconds, refers not only to the electron flow per second, but also the duration of flow.
One keV is equal to 1000 electron volt, and refers the energy of the X-ray photons. It is the unit of energy used in atoms and radiations, since joule is a larger unit.
Heat unit is the product of applied potential difference, tube current and exposure time and is given by
It is used to describe the heat produced for a given exposure in X-ray tube.
PHYSICAL QUANTITIES
Velocity, Acceleration and Momentum
Displacement (d) is defined as the shortest distance between the initial and final positions of a body. The velocity (v) of a moving body is the rate of change of displacement of the body in a 6particular direction and its unit is ms−1. Velocity is a measure of how fast the matter is moving or the rate of change of its position with time. It is given by the relation:
where d is the displacement in t seconds. The magnitude of velocity is called speed, which is a scalar quantity.
Acceleration (a) is defined as the rate of change of velocity and its unit is ms−2. It is a measure of how quickly or slowly the velocity is changing. If the velocity is constant, the acceleration is zero. It is given by the relation
where vf is the final velocity and v0 is the initial velocity, that a matter undergoes during the time interval t. The momentum (P) of a moving body is the product of mass (m) and velocity (v) and it is given by the relation:
The momentum is a vector quantity and its direction is the same as its velocity, the unit is kg-ms−1.
Scalar and Vector Quantities
All physical quantities can be classified into two broad categories namely, scalar and vector quantities. Quantities that have only magnitude and no direction are called scalar quantities, e.g. length, mass, time, etc. Quantities that have magnitude as well as direction are called vector quantities, e.g. displacement, velocity, force, etc.
A vector quantity is usually represented by a line with an arrow head (→). The magnitude of the vector is shown by the length of the line and the direction of the arrow represents the direction of the quantity. To find the resultant of scalar quantities, they are simply added algebraically. To find the resultant of two or more vector quantities, simple algebraic addition is not applicable. The addition of vector is done by the parallelogram law or the polygon law of vectors.
Force
Force is the influence that changes or tends to change the state of rest or of uniform motion of a body along a straight line. Let force F acts on a body of mass m and produces an acceleration of a, then
Hence, the force acting on the body is equal to the product of mass of the body and the acceleration produced by the force on the body. This is the Newton's second law of motion.
The SI unit of force is newton and it is denoted by the letter N. One newton is the force acting on a body of mass one kilogram producing an acceleration of one meter per second2 in its direction.
Pressure
The total force acting on a liquid surface is called thrust. The pressure (p) is defined as the force (F) per unit area (A) and its unit is Nm−2 or pascal (Pa). The atmospheric pressure is about 1.01×105 Pa. The pressure is caused by the weight of material pressing on its surface. It may be also due to collisions of atoms or molecules of a gas within a container. The pressure of a liquid at rest is always perpendicular to the pressure of the surface in contact with it. The pressure at 7a point within a liquid is directly proportional to the depth of the point from the free surface, density, and acceleration due gravity.
Gravitational Force
The force that pulls a body downwards with an acceleration is called gravitational force. The above acceleration is called acceleration due to gravity (g) and is equal to 9.81 ms−2. It is same for bodies of different masses and obey the relationship F= ma.
Mass and Weight
Mass is the amount of matter in a body and is expressed in kilograms. Weight is the downward force acting on the body, due to earth's gravitational attraction and is expressed in newtons. The relation between weight and mass can be written as W = mg, where, W is the force acting on the body called weight, m is the mass of the body and g is the acceleration due to gravity. Mass is always present in the body, but weight arises only in the presence of gravitational field. Hence, in space, the human body loses its weight.
Work
If a force acts on a body and the point of application of the force moves, then work is said to be done by the force. If the force F moves a body through a distance(s) in its direction, then the work done by the force is given by
The displacement does not always take place in the direction of force. If a constant force F applied on a body produces a displacement s in such a way that s is inclined to F by an angle θ, then the work done,
Hence, the work done is equal to the product of the component of the force in the direction of motion and the distance traveled. The SI unit of work is joule, and it is denoted by the letter J. One joule is the amount of work done, when the point of application of force of one newton acting on a body, moves it through a distance of one meter in the direction of force (1J = Ns).
Power
The rate of doing work is called power. It is measured by the amount of work done in unit time. If W is the work done in time t, then power
The SI unit of power is joule per second. It is also given by a special unit watt, which is equal to 1 joule per second. A larger unit of power is called kilowatt, which is equal to 1000 watt. The unit of electrical energy consumption is kilowatt-hour (Kwh). One kilowatt-hour is the power consumed at the rate of 1000 watts for one hour:
Energy
The Energy of a body is its ability to do work. It is measured by the amount of work that it can perform. The SI unit of energy is joule. There are many forms of energy, such as mechanical energy, heat energy, light energy, electrical energy, chemical energy, atomic energy, etc. There are two forms of mechanical energy, viz potential energy and kinetic energy.
The potential energy of a body is the energy it possesses by virtue of its position or state of strain, e.g. water stored up in a reservoir, a wound spring, compressed air, etc. For a body of mass m remaining at rest at a height h above the ground, the potential energy is equal to the work done in raising the body from the ground to that height.
where, g is the acceleration due to gravity.
Example 1.1:
A patient of weight 50 kg on a wheelchair has to be lifted onto a examination couch, which is 25 cm higher than wheelchair. Calculate the work done to carry out the above task (g = 9.81 ms−2).
The work done in lifting the patient onto the couch needs 120 J energy, which increases, the potential energy of the patient.
Kinetic Energy
The kinetic energy of a body is the energy possessed by it on account of its motion, e.g. moving car, water flow, etc. For a body of mass m moving with a velocity v, the kinetic energy is given by,
Example 1.2:
A Film cassette of mass 2 kg is kept in a shelf at a height of 1.5 m, possess a potential energy of 25 J. If the cassette falls on to the floor, what will be its speed?
The cassette may fell on the floor with a speed of 5 ms−1.
Density
The density of a body (ρ) is defined as the ratio of its mass (m) and volume (v) and its unit is kgm−3. The density of a body is same, if it is made up of identical material. If its composition is changed, its density will vary.
The relative density or specific gravity of a substance is the ratio between its density with that of water.
Mole
The amount of matter in a body is expressed by the number of elementary particles (atoms or molecules) it contains and its unit is mole. One mole of matter contains 6.022 × 1023 elementary particles, and it is known as Avogadro's number.
Boyle's law states that the volume (V) of a given mass of gas is inversely proportional to its pressure (P), at constant temperature. Charles's law states that volume of a given mass of gas, at constant pressure is proportional to its temperature (T). The above two laws can be combined and stated as follows:
This is known as the perfect gas equation.
TEMPERATURE AND HEAT
Matter is made up of atoms or molecules. These atoms and molecules are in regular movement in solids and random movement in liquids and gases. Hence, they possess kinetic energy which is responsible for the hotness and coldness of the body. Temperature is the measure of hotness and coldness of the body. When a body is heated, its molecules are in vigorous movement, and therefore have high energy, and the body is said to be in high temperature. When a body is cooled gradually, its kinetic energy decreases, and the body is said to be in lower temperature. Temperature is measured in degrees with the help of thermometers. There are three scales of temperature, namely:
- Centigrade scale
- Absolute scale
- Fahrenheit scale.
Centigrade Scale
In the centigrade scale, the temperature of the melting point of ice is taken as 0°C and temperature of the boiling point of water is taken as 100°C. The interval between the two is divided into 100 degrees. The scale is also called as Celsius scale.
Absolute Scale
In the absolute scale, Absolute zero is denoted as 0 K. The absolute zero is the temperature at which the molecules will have zero speed. The temperature of 0 K is equal to -273°C in centigrade scale. The temperature of melting ice is taken as 273 K and the temperature of boiling water is taken as 373 K. The interval between the two is divided into 100 degrees. One degree interval is the same in both centigrade and absolute scale of temperature. The absolute scale is also known as Kelvin scale of temperature. In the SI system of units, the absolute scale is used.
Fahrenheit Scale
In this scale, the melting point of ice is at 32°F and boiling point of water is at 212°F. The entire range is divided into 180 degrees. The body temperature is about 98.4°F equal to 37°C or 310 K. The relation between Celsius and Fahrenheit scale is given by
Worked Example 1.3:
Convert 86°F into degrees of celsius
Here, F = 86
Heat is a form of energy, which can be transferred from one place to another. If a hot body and a cold body are placed in close contact, the hot body will transfer some of its heat energy to the cold body until the temperature of the two become equal.
There are three methods of heat transfer. They are conduction, convection and radiation.
Conduction
Conduction is the process in which heat energy is transferred without the visible motion of the particles of the heated body. Conduction takes place in solids, liquids and gases. Metals in general are good conductors of heat, e.g. silver, copper, etc. Nonmetals are bad conductors of heat, e.g. glass, rubber, wood, etc.
Let us consider a rod of length L and area A and temperature θ1 and θ2 at their ends. The rate of flow of heat (dQ/dt) is directly proportional to cross sectional area (A), temperature gradient (θ1 – θ2)/L and thermal conductivity (k) of the material. The thermal conductivity of a material is its inherent ability to conduct thermal energy and it is expressed in Wm−1K−1. The relation for thermal conductivity is given by
Metals in general are good conductors of heat, e.g. silver, copper etc. Nonmetals are bad conductors of heat, e.g. glass, rubber, wood, etc.
Convection
Convection is the process in which heat energy is transferred by the actual motion of the particles of the body. Heat in liquid causes the fluid to expand and making it less dense and starts rising. The cold, dense fluid molecules move to their place from other area. Convection takes place in liquids and gases, e.g. trade winds, land and sea breezes.
Radiation
Radiation is the process by which heat energy is transmitted from one place to another without the aid of any material medium. When a body has internal energy, its atoms and molecules vibrate and emits electromagnetic radiation, which can transport energy across a vacuum, e.g. heat reaches the earth from the sun.
A black body and matt surface will radiate and absorb energy efficiently, while white and glossy surface will not. Stefan's law states that the rate of heat energy emission (dQ/dt) is directly proportional to the area of the emitting surface (A) and the fourth power of its temperature (T) dQ
where, σ is the Stefan-Boltzmann constant = 5.670 × 10−8 W m−2K−4.
The SI unit of heat is joule, however the special unit calorie is still in use. One calorie is the amount of heat which will raise the temperature of one gram of water by one degree Celsius and 1 calorie = 4.2 joules.
Thermal Expansion, Evaporation and Vaporization
When a material is heated, its atoms and molecules gains kinetic energy, and makes the material to expand. Expansion can take place three dimensionally. The linear expansion is the fractional change in unit length per unit change of kelvin temperature.
11When a liquid is heated, some of the atoms at the surface gains kinetic energy and escape from the surface. This is known as evaporation. The kinetic energy so gained must be greater than the force of attraction offered by the neighboring atoms at the liquid surface.
Vaporization is the process of liberation of atoms from solid materials. If a solid is heated sufficiently, its atoms at the surface gains sufficient kinetic energy and escape from the surface. In general, the atoms in a solid experience higher force of attraction from their neighbors than in liquid. Hence, vaporization requires higher amount of heat than that required for evaporation. A good example is the vaporization of tungsten target in an X-ray tube, which require very higher amount of heat.
Heat Capacity
When a body is heated, there is an addition or removal of internal energy. The above change of internal energy is quantified by the heat capacity. The heat capacity (C) of a material is the change in internal energy per unit temperature. It is the ratio of internal energy change to the temperature raise. It is independent of material size or shape and expressed in JK−1. It is given by the relation:
where, Q is the internal energy and Δθ is the change in temperature.
Specific Heat Capacity
The specific heat capacity (c) is the change in internal energy per unit temperature per unit mass of the material. That is, it is the heat required to raise temperature of 1kg material by 1 K, and it is expressed in Jkg−1K−1. It is given by the relation:
where, Q is the internal energy, Δθ is the change in temperature and m is the mass of the body.
Latent Heat
Latent heat is the property by which a material can absorb or release energy without rise or fall of temperature. For example, a solid melts into liquid without any rise in temperature, though it absorbs energy. Similarly, a liquid freezes into a solid without fall in temperature, though it releases energy. The absorbed or released energy is used for the rearrangement of particles, instead of changing the kinetic energy. Since the energy is hidden, it is called latent heat.
The specific latent heat is the hidden energy per unit mass and its unit is Jkg−1.The specific latent heat of water is 330 kJ. That is, 330 kJ of energy is required to melt 1 kg of ice into water at the same temperature (0°C). The latent heat of fusion relates the water that freezes into an ice. The latent heat of vaporization relates the boiling of water into vapor.
RADIOLOGICAL MATHEMATICS
Base and Exponent
A physical quantity (a) is often written as either a1, or a2 or in general an. Here, a, is the base and n is the exponent or power. The term an, is often called an exponential term. It refers to repeated multiplication, provided the exponent is positive integer as follows;
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If n = 1/2, it is not a positive integer, then, 
If n = 0, then a0 = 1, provided, a ≠ 0
An exponent can have minus sign, i.e. 
Two exponentials can be multiplied or divided or raised to a power as follows;
Decimal and Binary Numbers
We generally use decimal numbers which has a base 10 and involves digits 0 to 9. In a binary numbers, the value of a digit in a position is 2 times than it is in right. In a decimal number it is 10 times than it is in right. Decimal point is allowed in decimal system, where as it is not allowed in binary. For example the value of a binary number 10 is written as follows;
Similarly, one can convert any binary number into a decimal number by raising powers of 2 in a series and then adding them. Table 1.3 gives the equivalence of binary and decimal numbers.
Digital memory and storage uses the term bit, bytes and words etc. A bit is a small portion of a disc or tap that can be magnetized for data storage. Bits are grouped in bytes and words and 1 byte (B) is equal to 8 bit. The number of bits in a word may be 16,or 32 or 64, depending upon the computer system. Normally, kilo bytes (210 =1024 byte), mega bytes (220 =1048 kB) and giga bytes (230 = 1073 MB) are commonly used.
Logarithms
The logarithm of a decimal number is the exponent to which the base must be raised to produce the number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, and is written as logb (x), so log10 (1000) = 3.
There are three types of logarithms, namely; common logarithm (log10), natural logarithm (loge), and binary logarithms (log2), where e = 2.71828, a constant and often called as irrational number, e.g. log10 2 = 0.301, the base 10 must raised to power of 0.301, 100.301 = 2. Similarly, loge 2 = 0.693, the base e must be raised to power of 0.693, e0.693 = 2.
The measurements of optical density and sound intensity are expressed in logarithm to base 10. Radioactive decay, and X-ray attenuation uses logarithm of base e, which is denoted by lne (natural logarithm). Logarithmic scales reduce wide-ranging quantities to smaller scale. Logarithm is useful to describe many radiation events, such as X-ray absorption, radioactive decay, etc.
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Graphs
Graph gives the relationship between physical quantities, plotted as series of points or lines with reference to the set of axis. A Cartesian graph has two axis, namely x-axis called abscissa and y-axis called ordinate. The x-axis contains independent variable (time, distance) and the y-axis contains dependent variable (velocity, exposure).
If a physical quantity y varies with x in a proportional way, then a linear plot can be drawn. It is a straight line graph obeying the equation
where m is the slope of the line and c is the intersection with the y-axis.
Logarithmic functions, such as ex and e−x can also be plotted as curve, where a rapid increase or rapid decrease of a variable may be seen. A semi-log graph is a way of visualizing such data that is changing with an exponential relationship. One axis is plotted on a logarithmic scale and the other in linear scale. On a semi-log graph, the spacing of the scale on the y-axis is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values to their log, and plotting the data on linear (lin-lin) scales. The term log-lin is used to describe a semi-log plot with a logarithmic scale on the y-axis, and a linear scale on the x-axis (Fig. 1.1).
This kind of plot is useful, when one of the variables being plotted covers a large range of values and the other has only a restricted range. The advantage being that, it can bring out features in the data, that would not easily be seen, if both variables had been plotted linearly. Semi-log plot requires only few measurements of the exponential function.
Trigonometry
Trigonometry is a mathematics which deals with triangles and the relation between angle and sides (Fig. 1.2). If one angle of a triangle is 90° and the other angle is known, then the third angle can be obtained easily, since the sum of the angles is 180°. The two acute angles therefore added up to 90°, they are said to be complementary angles. The shape of a triangle is determined by the angles. Once the angles are known, the ratios of the sides can be determined, regardless of the overall size of the triangle.
14If the length of one of the sides is known, the other two can be determined. These ratios are given by the following trigonometric functions of the known angle θ, where a, b and c refer to the lengths of the sides in the accompanying figure:
Sine function (sin) is defined as the ratio of the opposite side to the hypotenuse, i.e.
Cosine function (cos) is defined as the ratio of the adjacent side to the hypotenuse, i.e.
Tangent function (tan) is defined as the ratio of the opposite side to the adjacent leg, i.e.
The hypotenuse is the side opposite to the 90° angle in a right angle triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle θ. The adjacent leg is the other side that is adjacent to angle θ. The opposite side is the side that is opposite to angle θ. The terms perpendicular and base are sometimes used for the opposite and adjacent sides, respectively.
The following are the values of sin, cos and tan functions, respectively;
Similar Triangle
Similar triangles have same corresponding angles and shape, but different sizes. The two triangles in Figure 1.3 are said to be similar triangles, since one is just bigger version of the other, while retaining the shape. In this, the corresponding angles of the two triangles are equal, but the lengths of the corresponding sides are not equal.
In a similar triangle, the lengths of the sides are in proportion to one another or the side ratios are equal as given below:
This means that, if one side has a length which is double that of the corresponding side of the other triangle, then all the sides will be doubled, compared to the other triangle. Similar triangle concept is used to follow a divergent radiation beam and to calculate field size at a given distance. It has role in inverse square law too.
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Fourier Transform
Jean Baptise Joseph Fourier was a French mathematician of 18th century. The Fourier transform (FT) is one of many transforms in mathematics. It is basically a mathematical mechanism, which converts a function from the time domain into the frequency domain. The frequency can be angular or linear. It is useful to converts a signal from an imaging device into a diagnostic image, e.g. MRI. FT can be presented in terms of graphs of functions, so that one can see how the FT changes the shape of these graphs. The FT is unique, and the inverse of FT does also exist. For example, FT can change a square wave into a sine function and 1/FT (inverse) of sine function can yield a square wave.
Let us consider a signal, whose intensity is varying with time as shown in Figure 1.4A. The signal is basically a time domain in real space. The waveform's amplitude and phase varies with time and the frequency of signal is said to be one cycle per cm. The FT converts the signal into intensity vs frequency (Fig. 1.4B). Now the data is in frequency domain of Fourier space. The FT decodes the frequency, phase and amplitude variations, corresponding to the position and amplitude of the data in time domain. The unit of Fourier space is the inverse of the unit in real space. That is, if s (time) is the unit in real space, than the unit of Fourier space is s−1 (1/time). The quantity s−1 is often known as frequency, hence, the name frequency domain or spatial frequency domain. Frequency is expressed in cycles per second or Hertz. It is easy to work in the frequency domain, which can be easily converted back to the time domain.
The real space and Fourier space are two different representations of the same data. In a real space, it shows how the signal varies with time. In Fourier space, one can see how the signal varies with frequency or how many frequencies are there in the signal. FT does not change the information present in a function, and offers newer view points on the data.
Fourier transform can be performed either two dimensionally (2D) or three dimensionally (3D), which finds application in Magnetic resonance imaging. The 2D Fourier transform is applied in length and width of the image matrix. The 3D Fourier transform is applied for each column, row and depth axis in the image matrix cube. This concept is used in 3D image acquisition in MRI.