1. X-ray Production The following slides identify atomic structure, the forces at work inside the atom, types of electromagnetic radiation (including x-rays), x-ray characteristics, components of an x-ray machine and x-ray tube, how x-rays are formed and ways to modify the x-ray beam. 0
2. An atom is composed of electrons (with a negative charge), protons (with a positive charge) and neutrons (no charge). The protons and neutrons are found in the nucleus of the atom and the electrons rotate (orbit) around the nucleus. The number of electrons equals the number of protons in an atom so that the atom has no net charge (electrically neutral). Different materials (for example, gold and lead) will have different numbers of protons/electrons in their atoms. However, all the atoms in a given material will have the same number of electrons and protons. (See diagram next slide) 0 Atomic Structure
3. 0 Atom This atom has 7 protons and 7 neutrons in the nucleus. There are 7 electrons orbiting around the nucleus. protons neutrons electrons
4. The electrons are maintained in their orbits around the nucleus by two opposing forces. The first of these, known as electrostatic force, is the attraction between the negative electrons and the positive protons. This attraction causes the electrons to be pulled toward the protons in the nucleus. In order to keep the electrons from dropping into the nucleus, the other force, known as centrifugal force, pulls the electrons away. The balance between these two forces keeps the electrons in orbit. (See next three slides) 0
5. Electrostatic force is the attraction between the positive protons and negative electrons. Electrons in the orbit closest to the nucleus (the K-shell) will have a greater electrostatic force than will electrons in orbits further from the nucleus. Another term often used is binding energy ; this basically represents the amount of energy required to overcome the electrostatic force to remove an electron from its orbit. For our purposes, electrostatic force and binding energy are the same. The higher the atomic number of an atom (more protons), the higher the electrostatic force will be for all electrons in that atom. 0
7. The balance between electrostatic force and centrifugal force keeps the electrons in orbit around the nucleus 0 EF CF
8. Electromagnetic Radiation An x-ray is one type of electromagnetic radiation. Electromagnetic radiation represents the movement of energy through space as a combination of electric and magnetic fields. All types of electromagnetic radiation, which also includes radiowaves, tv waves, visible light, microwaves and gamma rays, travel at the speed of light (186,000 miles per second). They travel through space in wave form. 0
9. D W W 0 The waves of electromagnetic radiation have two basic properties: wavelength and frequency. The wavelength (W) is the distance from the crest of one wave to the crest of the next wave. The frequency (F) is the number of waves in a given distance (D). If the distance between waves decreases (W becomes shorter), the frequency will increase. The top wave above has a shorter wavelength and a higher frequency than the wave below it. F = 3 F = 2
10. radio waves tv waves visible light x-rays gamma rays cosmic rays 0 Which of the above examples of electromagnetic radiation has the shortest wavelength? Which of the above has the lowest frequency? Cosmic rays Radio waves
11. 0 The energy of a wave of electromagnetic radiation represents the ability to penetrate an object. The higher the energy, the more easily the wave will pass through the object. The shorter the wavelength, the greater the energy will be and the higher the frequency, the greater the energy will be. X-ray Energy
12. A B C 0 Which of the above x-rays has the highest energy? A: It has the shortest wavelength, highest frequency
13.
14.
15. X-ray Equipment 0 X-ray equipment has three basic components: (1) the x-ray tubehead, which produces the x-rays, (2) support arms, which allow you to move the tubehead around the patient’s head and (3) the control panel, which allows you to alter the duration of the x-ray beam (exposure time) and, on some x-ray machines, the intensity (energy) of the x-ray beam. 1 3 2
16. 0 PID (cone) X-ray Tubehead degrees The x-ray tubehead is attached to the support arms so that it can rotate up and down (vertically;measured in degrees) and sideways (horizontally) to facilitate proper alignment of the x-ray beam. The PID (Position Indicating Device) is attached to the x-ray tubehead where the x-ray beam exits and it identifies the location of the x-ray beam. Some people refer to the PID as a “cone”; the PID’s on very old x-ray machines used to be coneshaped.
17. 0 The control panel, like the one above left, allows you to change exposure time but nothing else. Some machines, like the one above right, have controls for changing the mA and kVp settings in addition to exposure time. The individual controls will be discussed more later. exposure time kVp control mA control
18. X-ray Tube 0 X-rays are produced in the x-ray tube, which is located in the x-ray tubehead. X-rays are generated when electrons from the filament cross the tube and interact with the target. The two main components of the x-ray tube are the cathode and the anode.
19. (tungsten) Cathode Focusing cup Filament 0 The cathode is composed of a tungsten filament which is centered in a focusing cup. Electrons are produced by the filament (see next slide) and are focused on the target of the anode where the x-rays are produced. The focusing cup has a negative charge, like the electrons, and this helps direct the electrons to the target (“focuses” them; electrons can be focused, x-rays cannot). side view (cross-section) front view (facing target)
20. Thermionic Emission 0 When you depress the exposure button, electricity flows through the filament in the cathode, causing it to get hot. The hot filament then releases electrons which surround the filament (thermionic emission). The hotter the filament gets, the greater the number of electrons that are released. (Click to depress exposure button and heat filament). x-section of filament hot filament electrons
21. Anode Copper stem Target 0 The anode in the x-ray tube is composed of a tungsten target embedded in a copper stem. When electrons from the filament enter the target and generate x-rays, a lot of heat is produced. The copper helps to take some of the heat away from the target so that it doesn’t get too hot. side view front view Target
22. The sharpness (detail) of images seen on a radiograph is influenced by the size of the focal spot (area in the target where x-rays are produced). The smaller the focal spot (target), the sharper the image of the teeth will be. During x-ray production, a lot of heat is generated. If the target is too small, it will overheat and burn up. In order to get a small focal spot, while maintaining an adequately large target , the line focus principle is used. 0 Line Focus Principle
23. Line Focus Principle Apparent (effective) focal spot size Actual focal spot size Target (Anode) Cathode PID 0 The target is at an angle (not perpendicular) to the electron beam from the filament (see above). Because of this angle, the x-rays that exit through the PID “appear” to come from a smaller focal spot (see next slide). Even though the actual focal spot (target) size is larger (to withstand heat buildup), the smaller size of the apparent focal spot provides the sharper image needed for a proper diagnosis.
24. Line Focus Principle The target is at an angle to the electron beam. If you looked up through the PID at this angled target, it would “appear” to be smaller, as seen above. Click to rotate target and see altered size (indicated by yellow dotted lines). Actual focal spot size (looking perpendicular to the target surface; see previous slide); the length is indicated by the white dotted lines below. 0 Looking up at target through open end of PID PID
26. 1. Focusing cup: focuses electrons on target 2. Filament: releases electrons when heated 3. Electron stream: electrons cross from filament to target during length of exposure 4. Vacuum: no air or gases inside x-ray tube that might interact with electrons crossing tube 5. Target: x-rays produced when electrons strike target 6. Copper stem: helps remove heat from target 7. Leaded glass: Keeps x-rays from exiting tube in wrong direction 8. X-rays produced in target are emitted in all directions 9. Beryllium window: this non-leaded glass allows x-rays to pass through. The PID would be located directly in line with this window. X-ray Tube Components (continued) 0
27. Target Beryllium Window Focusing cup (filament located inside) Photo of an X-ray Tube 0 Leaded glass
28. X-ray Machine Components Control Panel X-ray Tubehead 110, 220 line Timer Exposure switch mA selector kVp selector Autotransformer Step-down transformer Step-up transformer X-ray Tube Wires Oil 0
29. The x-ray machine is plugged into a 110-volt outlet (most machines) or a 220-volt outlet (some extraoral machines). The current flowing from these outlets is 60-cycle alternating current. Each cycle is composed of a positive and negative phase. X-rays are only produced during the positive phase; the target needs to be positive to attract the negative electrons from the filament. During the positive portion of the cycle, the voltage starts out at zero and climbs to the maximum voltage before dropping back down to zero and entering the negative phase. Each complete cycle lasts 1/60 of a second; there are 60 cycles per second. (See next slide) X-ray Machine Voltage 0
30. + 110, 220 - 110, 220 0 positive negative target positive; electrons flow target negative; no electron flow target positive; electrons flow 0 voltage starts at zero and reaches a maximum of 110 or 220 before going back to zero
31. Direct Current (Constant Potential) 60-cycle Alternating Current 0 Many machines now convert the alternating current into a direct current (constant potential). Instead of cycles going from zero to the maximum, both positive and negative, the voltage stays at the maximum positive value, creating more effective x-ray production. This allows for shorter exposure times.
32. Timer 0 The timer controls the length of the exposure. The black numbers above represent impulses. The red numbers are seconds.
33. Number of Impulses 60 = seconds 1/60 sec. 0 With alternating current, there are 60 complete cycles each second; each cycle represents an impulse and is 1/60 of a second. To change impulses into seconds, divide the number of impulses by 60. To convert seconds to impulses, multiply by 60. Number of seconds X 60 = impulses 60 impulses/60 = 1 second 30 impulses/60 = 0.5 (1/2) second 15 impulses/60 = 0.25 (1/4) second 0.75 (3/4) second X 60 = 45 impulses 0.1 (1/10) second X 60 = 6 impulses
34. There are two electrical circuits operating during an x-ray exposure. The first of these is the low-voltage circuit that controls the heating of the filament. When the exposure button is depressed, this low voltage circuit operates for ½ second or less to heat up the filament. There are no x-rays produced during this time. As you continue to depress the exposure button, the high-voltage circuit is activated. This circuit controls the flow of electrons across the x-ray tube; during the positive portion of the alternating current cycle, the negative electrons are pulled across the x-ray tube to the positive target. X-rays are produced until the exposure time ends. The length of time the high-voltage circuit is operating represents the exposure time. 0
35. Exposure Button 0 The timer determines the length of the exposure, not how long you hold down the exposure button; you cannot overexpose by holding the exposure button down for an extended period. However, you can underexpose by releasing the exposure button too soon; the exposure terminates as soon as you release the button.
36. mA setting milliAmpere (mA) selector 0 The mA (milliAmpere) setting determines the amount of current that will flow through the filament in the cathode. This filament is very thin; it doesn’t take much current (voltage) to make it very hot. The higher the mA setting, the higher the filament temperature and the greater the number of electrons that are produced.
37. Step-Down Transformer 0 If the voltage flowing through the filament is too high, the filament will burn up. In order to reduce the voltage, the current flows through a step-down transformer before reaching the filament. The voltage reaching the step-down transformer is determined by the mA setting. The step-down transformer reduces the incoming voltage to about 10 volts, which results in a current of 4-5 amps flowing through the filament.
38. Step-Down Transformer Primary Secondary 0 The current enters the step-down transformer on the primary (input) side and exits on the secondary (output) side. The fewer turns in the coil on the secondary side, the lower the output voltage will be. The primary coil below would have 110 turns, the secondary coil would have 10. (Each loop of the coil is a “turn”; the number of turns in the diagram below has been reduced for easier viewing). 110 volts or less current flow 10 volts current flow
39. kiloVolt peak (kVp) control kVp readout kVp control knob The kVp control regulates the voltage across the x-ray tube. (A kilovolt represents 1000 volts; 70 kV equals 70,000 volts. A 70 kVp setting means the peak, or maximum voltage, is 70,000 volts). The higher the voltage, the faster the electrons will travel from the filament to the target. The kVp control knob regulates the autotransformer (see next slide). 0
40. Autotransformer The autotransformer determines how much voltage will go to the step-up transformer. Basically, a transformer is a series of wire coils. In the autotransformer, the more turns of the coil that are selected (using the kVp control knob), the higher the voltage across the x-ray tube will be. This is similar to the function of a rheostat. The following slide shows how this works. The incoming line voltage will be 110 volts. The exiting voltage will be 65 volts if the kVp control is set at 65. The exiting voltage will be 80 volts if the kVp setting is 80. 0
41. 110 V 65 volts current flow Autotransformer: the initial setting is 65; 65 volts leave the autotransformer. 0 80 volts to step-up transformer kVp selector Autotransformer:
42. The voltage coming from the autotransformer next passes through the step-up transformer, where it is dramatically increased. The ultimate voltage coming from the step-up transformer is roughly a thousand times more than the entering voltage. For example, if you set the kVp control knob to 65, 65 volts will exit the autotransformer. This 65 volts is increased to 65,000 volts by the step-up transformer. (The “k” in kVp stands for one thousand; 65 kV is 65,000 volts). The side of the step-up transformer where the voltage enters (primary side) has far fewer turns in the coil than the exit (secondary) side. 0 Step-Up Transformer
43. Step-Up Transformer Primary Secondary 0 The current enters the step-down transformer on the primary (input) side and exits on the secondary (output) side. The more turns in the coil on the secondary side, the higher the output voltage will be. The secondary coil in the step-up transformer has 1000 times as many turns as the primary coil. (Again, the number of turns has been reduced for easier viewing). 65-90 volts current flow 65,000 to 90,000 volts current flow
44. 0 65,000 to 90,000 volts 110 volts 10 volts The relationship of the various x-ray machine components are shown in the diagram below. They form the high-voltage and low-voltage circuits. For a more detailed review of the components, see next slide. kVp filament
45. oil filter The x-ray machine is plugged into the electrical outlet (110 volts usually). filament 0 exposure button
46. 0 The tubehead is filled with oil which surrounds the transformers, x-ray tube and electrical wires. The primary function of the oil is to insulate the electrical components. It also helps to cool the anode and, as we will discuss later, it helps in filtration of the x-ray beam. The barrier material prevents the oil from leaking out of the tubehead but still allows most x-rays to pass through. oil barrier material Step-up Trans Step-down Trans
47. X-ray Production 0 There are two types of x-rays produced in the target of the x-ray tube. The majority are called Bremmstrahlung radiation and the others are called Characteristic radiation.
48. Bremmstrahlung x-rays are produced when high-speed electrons from the filament are slowed down as they pass close to, or strike, the nuclei of the target atoms. The closer the electrons are to the nucleus, the more they will be slowed down. The higher the speed of the electrons crossing the target, the higher the average energy of the x-rays produced. The electrons may interact with several target atoms before losing all of their energy. Bremsstrahlung Radiation (Also known as braking radiation or general radiation) 0
49. Bremsstrahlung X-ray Production + 0 High-speed electron from filament enters tungsten atom Electron slowed down by positive charge of nucelus; energy released in form of x-ray Electron continues on in different direction to interact with other atoms until all of its energy is lost
50. Bremsstrahlung X-ray Production Maximum energy 0 High-speed electron from filament enters tungsten atom and strikes target, losing all its energy and disappearing The x-ray produced has energy equal to the energy of the high-speed electron; this is the maximum energy possible +
51. Characteristic Radiation 0 Characteristic x-rays are produced when a high-speed electron from the filament collides with an electron in one of the orbits of a target atom; the electron is knocked out of its orbit, creating a void (open space). This space is immediately filled by an electron from an outer orbit. When the electron drops into the open space, energy is released in the form of a characteristic x-ray. The energy of the high-speed electron must be higher than the binding energy of the target electron with which it interacts in order to eject the target electron. Both electrons leave the atom.
52. Characteristic x-rays have energies “characteristic” of the target material. The energy will equal the difference between the binding energies of the target electrons involved. For example, if a K-shell electron is ejected and an L-shell electron drops into the space, the energy of the x-ray will be equal to the difference in binding energies between the K- and L-shells. The binding energies are different for each type of material; it is dependent on the number of protons in the nucleus (the atomic number). Characteristic Radiation (continued) K-shell M-shell L-shell 0
53. Characteristic X-ray Production L K M High-speed electron with at least 70 keV of energy (must be more than the binding energy of k-shell Tungsten atom) strikes electron in the K shell, knocking it out of its orbit Ejected electron leaves atom Recoil electron (with very little energy) exits atom 0 X-ray with 59 keV of energy produced. 70 (binding energy of K-shell electron) minus 11 (binding energy of L-shell electron) = 59. vacancy Electron in L-shell drops down to fill vacancy in K-shell
54. X-ray Spectrum An x-ray beam will have a wide range of x-ray energies; this is called an x-ray spectrum. The average energy of the beam will be approximately 1/3 of the maximum energy. The maximum energy is determined by the kVp setting. If the kVp is 90, the maximum energy is 90 keV (90,000 electron volts); the average energy will be 30. As shown below, characteristic x-rays contribute a very small number of x-rays to the spectrum. X-ray energy (keV) characteristic x-rays (59 & 67 keV) # of x-rays 0 Bremmstrahlung x-rays
55.
56. 0 X-ray production is a very inefficient process. Only 1% of the interactions between the high-speed electrons and the target atoms result in x-rays. 99 % of the interactions result in heat production. The excess heat is controlled by the high melting point of the tungsten target, the conductive properties of the copper sleeve and the cooling from the oil surrounding the x-ray tube. heat
57. X-ray Beam Modifiers 0 The following slides identify the various ways of changing the energy of the x-ray beam and the number of x-rays produced during an x-ray exposure.
58. Exposure Factors 0 The energy of the x-ray beam and the number of x-rays are primarily regulated by the kVp control, the mA setting and the exposure time. One, two or all three of these exposure factors may need to be adjusted, depending on the size of the patient’s head, the likelihood of patient movement due to tremors or the inability to hold still, etc.. If the exposure factors are not set properly for the current patient, the resultant film may be too light or too dark (see next slide).
59. Exposure factors too high (too dark) Correct exposure factors Exposure factors too low (too light) 0
60. kVp (kilovolt peak) 0 The kVp primarily controls the energy or penetrating quality of the x-ray beam. The higher the kVp, the higher the maximum energy and the higher the average energy of the beam. A higher kVp allows the x-ray beam to pass through more dense tissue in a larger individual, resulting in a more acceptable radiographic image. In addition to increasing penetrating ability, a higher kVp will also result in the production of more x-rays. Because of this, an increase in kVp will allow for a decrease in exposure time, which may be helpful in children or in adults with uncontrolled head movement.
61. kVp (kiloVolt peak) 0 X-ray Energy (keV) Number of X-rays 70 90 90 kVp 70 kVp In switching from 70 kVp to 90 kVp, the average energy increases (dotted lines below), the maximum energy increases (from 70 keV to 90 keV) and the number of x-rays increases. (Click to change from 70 kVp to 90 kVp).
62. mA (milliampere) 0 The mA setting determines the heating of the filament. The hotter the filament, the more electrons that are emitted; the more electrons crossing the x-ray tube, the greater the number of x-rays that result. There is no change in the average energy or maximum energy of the x-ray beam. Doubling the mA setting results in twice as many x-rays. (Click to change from 5 mA to 10 mA). Number of X-rays X-ray Energy 10 mA (twice as many x-rays) 5 mA maximum energy average energy (no change) (no change)
63. Number of X-rays X-ray Energy 10 impulses (twice as many x-rays) 5 impulses maximum energy average energy 0 Exposure Time An increase in exposure time will result in an increase in the number of x-rays. Doubling the exposure time doubles the number of x-rays produced. Exposure time has no effect on the average or maximum energy of the x-ray beam. (Click to change exposure time from 5 impulses to 10 impulses). (no change) (no change)
64. mAs or mAi 0 mAs = milliamperes (mA) x seconds (s) mAi = milliamperes (mA) x impulses (i) All x-ray machines have an mA setting (may be fixed or variable) and an exposure time setting (always variable) for each radiograph taken. The product of the mA setting times the exposure time equals mAs or mAi, depending on whether the exposure time is in seconds or impulses. As long as the mAs remains constant for a given patient size, the x-ray output will remain the same. For example, if the mA setting is 5 and the exposure time is 30 impulses, the mAi would be 150 (5 times 30). If we change the mA setting to 10 and decrease the exposure time to 15, the mAi is still 150 (10 times 15). There will be no change in the number of x-rays. If an x-ray machine has variable mA settings, increasing the mA will allow for a decrease in exposure time; this will be advantageous in most cases.
65.
66. Filtration Low-energy x-rays do not contribute to the formation of an x-ray image; all they do is expose the body to radiation. Therefore, we need to get rid of them. The process of removing these low-energy x-rays from the x-ray beam is known as filtration. Filtration increases the average energy (quality) of the x-ray beam. There are two components to x-ray filtration. The first of these, called inherent filtration , results from the materials present in the x-ray machine that the x-rays have to pass through. These include the beryllium window of the x-ray tube, the oil in the tubehead and the barrier material that keeps the oil from leaking out of the tubehead. This removes very weak x-rays. 0
67. Filtration (continued) The second component is the addition of aluminum disks placed in the path of the x-ray beam ( added filtration ). These disks remove the x-rays that had enough energy to get through the inherent filtration but are still not energetic enough to contribute to image formation. Disks of varying thicknesses, when combined with the inherent filtration, produce the total filtration for the x-ray machine. Federal regulations require that an x-ray machine capable of operating at 70 kVp or higher must have total filtration of 2.5 mm aluminum equivalent. (The inherent filtration is “equivalent” to a certain thickness of aluminum). X-ray machines operating below 70 kVp need to have a total filtration of 1.5 mm aluminum equivalent. 0
68. Filtration Inherent beryllium window of x-ray tube Added Aluminum filter (s) Total Oil/Metal barrier 0 filter PID collimator barrier material beryllium window oil
69. filter PID The filter is usually located in the end of the PID which attaches to the tubehead. 0
70. primary x-ray scattered x-ray 0 Collimation Collimation is used to restrict the area of the head that the x-rays will contact. We want to cover the entire film with the x-ray beam, but don’t want to overexpose the patient. Also, when x-rays from the tubehead interact with the tissues of the face, scatter radiation is produced (see below). This scatter radiation creates additional exposure of the patient and also decreases the quality of the x-ray image. (Scatter will be discussed in greater detail in the section on biological effects of x-rays).
71. Collimation 0 The collimator, located in the end of the PID where it attaches to the tubehead, is a lead disk with a hole in the middle (basically a lead washer). The size of the hole determines the ultimate size of the x-ray beam. The shape of the hole will determine the shape of the x-ray beam. You are looking up through the PID at the collimator (red arrows), which is a circular lead washer with a circular cutout in the middle. This will produce a round x-ray beam. The light gray area in the center is the aluminum filter.
72. The shape of the opening in the collimator determines the shape of the x-ray beam. The size of the opening determines the size of the beam at the end of the PID. PID’s come in varying lengths; longer PID’s have a smaller opening in the collimator. 0 round rectangular Collimation
73. The x-ray beam continues to spread out as you get further from the x-ray source (target). More surface is exposed on the exit side of the patient than the entrance side. By collimating the beam, less overall surface is exposed and as a result, less scatter radiation is produced. Both of these things reduce patient exposure. 2.75 inches (7 cm) is the maximum diameter of a circular beam or the maximum length of the long side of a rectangular beam at the end of the PID. collimator 0 target (x-ray source) Collimation collimated beam
74. If you switch from a 7 cm round PID to a 6 cm round PID, the patient receives 25% less radiation because the area covered by the beam is reduced by 25%. Rectangular collimation (dotted line at left) results in the patient receiving 55 % less radiation when compared to what they would receive with a 7 cm round PID. 6 cm round film (4.5 cm long) entrance entrance exit exit 6 cm 7 cm 7 cm round 0 area covered at skin surface (6 cm round PID) area covered as beam exits (6 cm round PID) area covered at skin surface (7 cm round PID) area covered as beam exits (7 cm round PID) Collimation
75. Quality Quantity (primarily) kVp mA Time Filtration no change no change 0 Collimation does not change the energy or number of x-rays in the x-ray beam that reach the film; it just limits the size and shape of the beam. The quality, or average energy, of the x-ray beam is increased with an increase in kVp or an increase in filtration. The quantity, or number of x-rays, is increased with an increase in kVp, mA setting and kVp setting.
76. Inverse Square Law 0 The x-ray beam spreads out as it moves away from the target (focal spot, source, focus), covering a larger area and diluting the effects of the x-ray beam in a given area. The farther you get from the target , the weaker the x-ray beam will be. The inverse square law is a formula used to identify the strength (intensity) of the x-ray beam at a given distance from the target. (See next slide).
77. The formula above shows that the intensity of radiation varies inversely as the square of the target-film distance. D represents distance, KI is the known intensity (x-ray beam strength) and UI is the unknown intensity. If you know the intensity at one distance, you can compute the intensity at any other distance by using the formula. (See example on following slide). Inverse Square Law
78. D D KI UI 2 X KI = UI 1* 100 X 100 = 1 Inverse Square Law 0 The intensity of the beam at 1 ft. is 100 (KI). What is the intensity of the beam at 10 ft. (UI)? The distance of known intensity (D KI ) is 1. The distance with unknown intensity (D UI ) is 10. Using the formula, as seen at right, the unknown intensity is found to be 1. * 1 10 2 = 1 100
79. 0 Inverse Square Law The inverse square law also works by looking at the magnitude of change in the distance. In the diagram below, the distance D 2 is 2 times the distance of D 1 . The x-ray beam covers 4 squares at D 1 and 16 squares at D 2 , or four times as many; the intensity is ¼ as much because the beam is spread out over four times as many squares. The change in distance is two times; the square of 2 is 4 and the inverse of 4 is ¼. If the change in distance is 3 times as much, calculate as follows: the square of 3 is 9, and the inverse is 1/9. The intensity at three times the distance would be 1/9 what it was at the initial distance.
