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# Algebra PHYS 2020

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This 17 page Class Notes was uploaded by Vernie Wehner on Wednesday October 21, 2015. The Class Notes belongs to PHYS 2020 at Tennessee Tech University taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/225726/phys-2020-tennessee-tech-university in Physics 2 at Tennessee Tech University.

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Notes for Phys 2020 Last revised 42504 These notes are just intended to give an overview of the topics and major equations covered in class At present they lack illustrations which can be important for some topics 1 Electric Forces and Electric Fields 11 Electric Charge All the elementary particles in nature have a property a quantity known as electric charge Electric charge can be positive or negative in the 81 system electric charge is measured in coulombs Often we will let the letter Q stand for charge The electron carries a charge 6 and the proton carries a chargee where e 16022 gtlt10 19 C The nucleus of the atom contains the protons and also neutrons which have zero electric charge Usually an atom contains the same number of electrons as protons and so has a net charge of zero ie it is electrically neutral To see the effects of electric charge we need to separate the charges generally this means moving electrons from one place to another It turns out that for reasons not completely understood the free charges in nature must be some integer multiple of e that is q Ne where N is some integer Quarks can have charges of i3 and i275 but as far as anyone knows you cant isolate a single quark 12 Behavior of Charges Conductors are materials for which any excess charge is free to move through the material in response to electrical forces Insulators are materials for which an excess charge will stay where it is put 13 Coulomb s Law Electrical charges which have the same sign will repel one another by Newtonls Third Law they exert equal and opposite forces on one another which are directed away from the other charge Charges with opposite signs attract one another The forces are equal in magnitude and directed toward the other charge Suppose we have two charges with values ql and q2 They can be positive or negative charges Suppose the distance between the charges is 7 Then the magnitude of the force between them is given by Coulomb s law lQ1l lazl F k7 1 where k 899 gtlt 109 N512 To get the direction of the electric force on a given charge we need to think about the locations and the signs of the other charges When there are several other charges around a charge ql add up the individual force vectors to get the total electrical force on ql While the number k is useful in Coulombls law we will later nd it more useful to use a constant denoted as 60 The numbers are related by which gives 60 8854 gtlt 10 12 N95 i 4713960 14 The Electric Field Since by Coulombls law the force on any one charge is the sum of forces each of which is pmportianal to the given charge we will get a more useful and general quantity if we divide out the charge from that force when we do we get a quantity called the electric eld EF110 The electric eld is a vector and has units of NC For a given con guration of charges we can in principle nd the electric eld at any point When we do if we place a charge go at that point the force on that charge is Fq0E The electric eld form a point charge q at a distance of r from that charge has a magnitude E klriz lf q is positive E points away from the charge If q is negative it points toward the charge This gives a force which is consistent with Coulombls law The force due to an in nite uniform planar sheet of charge has magnitude E i where 039 is the surface charge density on the sheet This eld is directed away from the sheet if 039 is positive The electric eld in the region between two such uniform in nite sheets of opposite charge densities 0 and 70 has magnitude E g This eld is directed from the positive plate to the negative plate Often it is useful to make eld line diagrams to show the direction of the electric eld at all points in space When excess charge is placed on a conductor or when a conductor is placed in an external magnetic eld so that it becomes polarized the unbalanced charges all reside on the surface of the conductor Furthermore the electric eld is zero in the interior of the conductor Since the charges all move freely they will arrange themselves so that we have zero eld inside the interior 2 Electric Potential Energy and the Electric Potential When an electric charge moves through an electric eld the electric force does work on the charge The electric force is a conservative force so that we can talk about a potential energy due to electricityjust as we did with gravity rst semester An electric charge will thus tend to prefer locations where there is a low electric potential energy We will use EPE to denote this electric potential energy In fact7 for many of our problems we ill be able to ignore gravity and we will only talk about this kind of potential energy Electric potential energy is a scalar and is measured in joules Potential energy is a useful concept because we can use the principle of conservation of energy to solve many problems The total energy kinetic plus potential stays the same 21 The Electric Potential Since the force and the work done on a charge in moving it through an E eld are proportional to the charge7 it is useful to nd the electric potential energy per unit charge That is7 divide the charge in EPE by the charge to get a more useful quantity This quantity will be denoted by V and is called the electric potential So AV is given by i AEPE 110 AV The electric potential is a scalar and its units are JC We call this combination a Volt J 1 V 16 A useful unit of energy is the electroniVolt eV7 which is the change in EPE when a charge of magnitude 6 moves through a potential idfference of 1 Volt Speci cally7 1eV 1602 gtlt10 19 J For a point charge q7 when we are at a distance of r the electrical potential is V k9 r Donlt confuse this with a similar looking expression for the electric eld from a point charge Note that there is no absolute value in this expression it gives the scalar value of V To nd the electric potential at a particular location due to a group of point charges7 just add up the potentials due to the individual charges An equipotential surface is the set of points on which the electric potential has a certain value ie it is constant Electric eld lines are perpendicular to equipotential surfaces One can show that for a uniform electric eld7 the eld and potential are related by AV Ax that is7 the electric eld component in a certain direction is what you get by dividing the charge in V by the change in x for that direction with a minus sign out in front Piom this it is clear that the units of the electric eld can also be expressed as Vm Ex 3 22 Capacitors A capacitor as well use the term is a pair of conducting planes which are close together as compared with their dimensions Welll consider charges on these plates but only the case where there are opposite charges on the two plates q on one of them and 1 on the other The charge stored on the plates is proportional to the potential difference across the plates q CV where C is the capacitance of the device measured in Farads A Farad is equal to CV Coulomb per Volt A Farad is actually a huge amount of capacitance The capacitance of a pair of parallel plates of area A and separation d is A O 60E When we ll the area between the plates of a capacitor with an insulator we get a device with a capacitance which is larger by a factor of K where K is called the dielectric constant of the insulating material Thus 0 KOajr The dielectric becomes polarized when it is inside a charged capacitor that it the positive and negative charges have small displacements in opposite directions The effect of this polarization is for a xed charge on the plates q to decrease the electric eld inside the capacitor and also the voltage V This increases the capacitance by the factor K A capacitor stores electrical energy since it takes work to separate the charges If q and V are the charge and voltage on a capacitor C the energy stored is 1 2 12 E 7 20V 7 20 3 Electric Circuits 31 Electric Current Electric current for our purposes is the motion of charge within a conductor In reality it is the electrons which move in the conductor but because of a historical accident we will use the common convention that there are an equal number of pasz39tz39ve charges moving in the opposite direction For all of our work it wont make any difference Electric cuurent can be make to ow in a wire by setting up an electromotive force emf across its ends One can do this with a common battery for which there is a decrease in chemical energy is charge is delivered from one terminal of the battery to the other This causes the charges in the wire to do a slow drift so that the charge is transferred We observe the effects of electric current so quickly because the conductor is already lled with free charges For our purposes we will use emf77 voltage77 and potential difference77 interchangeably All of them are measured in volts Electric current is measured by counting the amount of charge that passes by any cross sectional area of the wire in a given amount of time If a charge q passes by some placein the wire in a time interval t then the current I is given by q 1 t Current is measured Cs coulombs per second7 which is abbreviated as the ampere 1A1 32 Ohrn s Law We nd that for most conductors the potential difference is proportional to the current passing through the conductor The constant of proportionality is the resistance of the conductor V IR This relation is known as Ohm s law Resistance has units of VA7 which is called an ohm and is abbreviated as 9 For a conductor with cross sectional A and length L7 the resistance is L R 7 p A The constant p is the resistivity of the material of which the resistor is made Resistivity has units of Q m 3 3 Alternating Current In fact the potential difference that is used to run household appliances ie the plug in the wall is not a constant value and if it is connected to a resistance the current is not constant either The wall voltage oscillates sinusoidally with an amplitude of about 165 V and a frequency of 60 cycles per second In the USA7 that is I will skip all the material on alternating current in lecture7 but since we will be solving a few problems which use household voltages it is worth mentioning how it is that we can still do this We rst de ne the rootimeanisquare values of the voltage and current7 which are the peak values divided by the square root of two When we use the rms values it turns out that we can use the DC circuit equations V IR and P IV with correct results 34 Resistors in Series and in Parallel When resistors are combined in series we mean that they are joined end to end without any junctions occuring between the resistors In such a con guration the current in all the resistors is the same though in general the potential difference acroos each will be different If the individual values of the resistors are R1 R2 R3 then the set can be replaced by an equivalent resistance which si just equal to the sum ReqR1R2R3 By this we mean that with this replacement you get get the same current if the same voltage is applied Whe resistors are connected in parallel we mean that their ends are joined together In such a con guration the potential dz erenee across each of the resistors is the same though in general the currents in each will be different If the individual values of the resistors are R1 R2 R3 then the equivalent resistance of the set is given by the relation Req R1 R2 With this replacement7 we get the same total current going into the resistor combination if the same voltage is applied across it 3 5 Kirchhoff s Rules To solve the really complicated DC circuits we need more than the rules for reducing sets of resistors By using two rules and lots 07 math one can solve for the currents in all the branches of a complicated circuit 0 At any junction7 the sum of currents entering the junction equals the sum of currents coming out of the junction 0 Around any loop the sum of voltage rises equals the sum of the voltage drops7 ie the sum of all the potential differences is zero 4 Magnetic Forces and Magnetic Fields 41 The Magnetic Field Permanent magnets have been known for hundreds of years7 ever since the ancient Greeks used them as a way to make their refrigerator doors more cheerful When a small magnet is free to rotate we nd that one end of it wants to point toward the geopgraphic North direction and the other to the South Appropriately these are called the North and South ends of the magnet One nds that the North ends of magnets or their South ends will repel one another It seems that every magnet has a North and South pole it is impossible to isolate Northness from Southness as one can do with positive and negative electric charges 42 Magnetic Forces A magnetic eld exerts a force on a particle of charge q but only if the charge is in motion and then only if the velocity of the particle has a component perpendicular to the direction of B If the velocity of the particle is completely perpendicular to B then the force on the particle has magnitude F qu and the direction of the force is perpendicular to both V and B Even having said this there is an ambiguity in the direction of the force7 but it is decided by the right hand rule which7 as well use it in this class7 goes as follows using your right hand 0 Thumb goes in direction of velocity 0 Fingers go in direction of magnetic eld 0 Palm faces in the direction of the force on the particle This gives the direction of the force on a moving positive charge lfthe charge is negative7 take the opposite direction from what this rule gives If the velocity of the particle is not perpendicular to the B eld and the angle between the velocity vector and the B eld is 9 then the magnitude of the force is given by F lquB sin0l 43 Motion of Particles in a Magnetic Field The fact that the force is peppendieiilm to the direction of motion of the charge has some interesting consequences When we have and E eld and a B eld at right angles to one another a charged particle passing through this region with a velocity perpendicular to both elds will experience no net force only if the speed of particle has the value 7 EB When a charged particle enters a region of a uniform magnetic eld with a velocity perpendicular to the eld direction7 the particle will move in a circular path The relation between the parameters of the motion is 77717 riqu where r is the radius of the orbit7 m and q are the mass and charge of the particle7 7 is the particles speed which does not change and b is the magnitude of the B eld Such a particle motion occurs in the accelerators of physics and in the mass spectromemter a device used in chemical analysis For these applications we are generally given the potential V through which the charge q is accelerated instead of its speed 71 Using qV mvz we nd that the relation between the parameters is 2mV 1 Br 44 Force On a CurrentiCarrying Wire If a length L of a straight wire carries a current I in a direction which is not parallel to the B eld it will experience a force which is perpendicular to both the direction of the current and the B eld The magnitude of this force is F ILBsint9 where 9 is the angle between the current direction and that of the B eld Again there is an ambiguity in the direction of the force which is resolved by the right hand rule as applied to currents7 0 Thumb goes in direction of current 0 Fingers go in direction of magnetic eld 0 Palm faces in the direction of the force on the wire 45 Magnetic Fields from Currents The magnetic eld around a long straight wire is given by M B 7 27139 where I is the current in the wire7 r is the distance from the wire and no is a new constant which appears in expressions where we nd the magnetic eld due to a current7 namely HO 47TX10 7T39TH The direction of the B eld is found from the second right hand rule7 in which the thumb of the right hand points in the direction of the current and the ngers wrap around the wire to give the direction of the B eld We can now nd the force between two very long straight wires which are parallel to one another If one wire carries a current 1 and the other a current 2 and they are separated by a distance r the force felt by a length L of one of the wires is F LH01112 27139 We nd that if the currents go in the same direction there is an attractive force and if the currents go in opposite directions the force is repulsive contrary to what one might thinkl Two other current con gurations are of interest At the center of a circular loop of radius R which carries a current I7 the magnitude of the magnetic eld is 13 B 2B A solenoid is a helix formed from a current carrying wire For a long solenoid the only kind welll consider7 the magnetic eld inside the helix is very nearly uniform7 and the B eld outside is weak If the number of turns per unit length is n and the current in the wire is I7 the magnitude of the B eld inside the solenoid is B Mon 5 Electromagnetic Induction 51 Changing Magnetic Field Makes an Emf We can induce an electric current to ow through the wire in a coil if we change the magnetic eld passing through the coil This will also happen if the magnetic eld through the coil stays the same but the shape area of the coil changes The current ows only while a change in the B eld occurs When the B eld through the coil is steady there is no current This is shown with a coil connect to an ammeter around which we make rapid changes in the magnetic eld using a cow magnet Yes7 they really put those tings inside cows 52 Example A Simple Circuit We do a thought experiment where a simple circuit with a single resistance like a alight bulb has two long leads which maintain contact with a sliding segment segment of length L We will use this simple circuit to develop ideas about EM induction When the segment slides on the rails the bulb lights up This is because an emf S is developed in the loop7 so that if the bulb has resistance B there is a current 8R in the loop To clarify matters7 we consider the motion of the conducting bar through the magnetic eld by itself Noting that the free charges now have a velocity perpendicular to the magnetic eld7 there must be magnetic force on them along the length of the bar 53 Magnetic Flux Magnetic ux through a surface bounded by a conducting loop for a uniform magnetic eld is de ned as the narmal campzmem of the magnetic eld times the area of the loop ltlgt BAcosaS where b is the angle between the B eld and the normal to the loop 54 Faraday s Law When the magnetic ux through a loop is changing and only while it is changing there is an induced emf in the conducting loop For a single loop the magnitude of the induced emf is given my Altlgt 5 7 At But the complete law needs a couple extras We can have a coil of N loop all with same shape and area which enhances the emf Also7 as we will see7 the induced emf has a sense which is opposite that of the changing magnetic ux With this in mind7 Faradayls law is usually written as 8 i AltIgt 7 it While the above formula gives the magnitude of the average emf induced over the time of the changing ux7 the direction of the induced current is not so clear It is much clear when we use Lenz s law7 which states The polarity of the induced current is such that the induced magnetic ux opposes the original change in ux 5 5 Electric Generators And thus we arrive at the means by which electric power is made If we rotate a coil within a magnetic eld7 the magnetic ux is continually changing because of the changing directian of the B eld 56 Mutual Inductance and Selfilnductance The changing ux within one circuit can arise from the fact that a current in a nearly circuit is changing7 since that will give rise to a changing magnetic eld which is felt by the former We will say that a changing current is a primary circuit 1 gives rise to an induced emf in the secondary circuit More speci cally the induced emf in the secondary is pmpartz39anal t0 the rate of change of the current in the primary circuit Since the ux in circuit 2 is proportional the to current in 17 we can write N2ltIgt2 M12 where N2 is the number of turns in the secondary circuit coil The constant M depends on the geometry of the two circuits it is called the mutual inductance of the two circuits Then we can combine the two equations no 52 7N2 A and N2AltIgt2 MAI1 to get A11 52 ME In the same way a current in a coil gives a magnetic ux thmugh the call itself and so a changing current through a coil generates a backwards emf De ning the selfiinductance by NltIgt LI where N is the number of turns in the coil we can then write AI 5 7 7 At We can give a formula for the self inductance of a solenoid It is L MonzAz where as before 71 is the number of turns per unit length A is the cross sectional area of the coils and Z is the length of the solenoid A solenoid used in this fashion is often just referred to as an inductor with its inductance measured in Henrys Whe current is owing through an inductor there is energy stored in the magnetic eld that is set up inside the coil The reason that energy is stored is that when the current is building up from zero there is a backwards voltage resulting from the changes in the current So the current must go through a rise in potential and so energy is expended When a current I is owing through an inductor with inductance L the energy stored is 2 gm 6 Electromagnetic Waves 61 Waves Formed From the E and B Fields When Maxwell found the equations that fully uni ed electricity and magnetism he was able to predict that there would be wave solutions to the equations this means that oscillating electric and magnetic elds can propagate over long distance in space just as oscillations propagate along a string or a sound wave propagates through air In an electromagnetic wave travelling through a vacuum the E and B elds are both perpendicular to the direction of propagation Furthermore these elds are perpendicular to one another and instantaneously have magnitudes related by EcB Since the elds are perpendicular to the direction of propagation the EM wave is a transverse wave Maxwell was able to predict the speed of these waves from the equations and he found that the speed of EM waves in a vacuum is vEMmd 300 gtlt 108 g 1 This was well known to be the speed of light so that Maxwell could speculate that light was a particular kind of EM radiation and that there could be other types having longer and shorter wavelengths Nowadays the speed of light in vacuum which is not much different from the sped of light in air as well see is denoted by c and to a better approximation is c 299792 gtlt108 In fact the speed of light is de ned to be a certain number of meters per second and since the second has a precise de nition in terms of atomic properties this provides the working de nition of the meter EM waves differ from the waves studied in Phys 2010 in that they do not travel through any kind of elastic medium which distorts to form the wave7 though at one time it was thought that this was the case The fact that EM waves dont need such a medium led to the development of the Theory of Relativity Wavelength7 frequency and wave speed for electromagnetic waves have the same relation we had for waves through an elastic medium Afc EM waves carry energy The intensity of an EM wave is the amount of energy which passes through a given cross sectional area in a given amount of time The intensity is the energy per time7 per area7 and is measured in units of gmz Wm2 For an EM wave for which the amplitude of the electric eld is E0 and the amplitude of the magnietic eld is B0 E0 CBO is C S ceoEg Bg 62 Polarization Given the direction of propagation of an EM wave7 the electric eld can point in any of the directions in the perpendicular plane The magnetic eld is there too7 at right angles to E but we will just discuss the direction of E7 for simplicity Welll refer the the direction of E as the polarization of the EM wave Now in fact the light that we get from natural sources like the sun or lamps does nut have a de nite polarization it is a combination of the light of all polarizations it is unpolarized But it was found that some crystals can force the light they transmit to have a de nite polarization The sheets made of these substances are called polaroids When unpolarized light is incident on a polaroid7 the transmitted light has half of its original intensity S ESQ and is now polarized in the direction of the axis of the polaroid But if light which is already polarized is incident on a polaroid whose axis has a direction which differ by and angle 9 from the polarization direction of the incident light7 the transmitted light is polarized in the new direction and has an intensity which is cos20 times that of the incident light 3 30 cos219 This is usually called the Law of Malus 7 Re ection and Mirrors In this chapter and the next we will analyze the formation of images by optical devices mirrors and lenses To get our results we will nd what happens to light rays which are incident on these objects When light rays encounter most surfaces they are re ected back7 but at essentially random angles This is known as diffuse re ection When light encounters a very smooth metallic 12 surface then specular re ection takes place When we measure angles from the normal to the surface then for specular re ection a ray incident at 9 goes out at 9 with 9 9 71 Plane Mirrors A at planar mirror takes the light rays coming from an object and re ects them back so that when traced backwards they seem to be coming from a place behind the mirror They come back at the view just as if there was something of the same size and at an equal distance behind the mirror The ctional source of the light is called the image 72 Spherical Mirrors If we take a spherical shell of radius R with a very smooth surface and coat the outside or inside with a relective material then take a piece of the sphere we have a spherical mirror Such a piece of the sphere has a center of curvature C which is at a distance R from all points of the mirror We will consider a particular line along a radius to use in analyzing the location of objects and images This line is called the principal axis The point on the axis midway between C and the mirror if the focal point of the mirror for this axis We denote it by F and we say that its distance from the mirror is f with f R2 For an object near the principal axis we can locate the image by tracing several rays for which the geometry is simple Starting with some point of the object not on the axis draw the rays 0 A ray which comes in parallel to the axis goes out along a line which passes through the focal point 0 A ray which comes in on a line through the focal point goes out parallel to the axis 0 A ray which comes in on a line through the center of curvature goes out along the same line Using these three rays actually 2 are suf cient we can locate the images and say whether it is upright or inverted We nd that for a concave mirror when the object is between the focal point and the mirror the object is in back of the mirror and upright If the object is more distant than the focal point the image is m from of the mirror and is inverted For a convex mirror we nd that the image is always behind the mirror and is upright If the image is in front of the mirror it is a real image If it is behind the mirror it is a Virtual image The image will have its own size as well If the object has a height measured perpendic ular to the principal axis ho and the image has height h then the magni cation by the mirror is m Note that when the image is inverted ho and h have opposite signs 73 The Mirror Equations To use the equation which gives the location we must be very careful with the signs of the distances involved Generally a length measured in fmm of the mirror is positive and one measured behind the mirror is negative In particular 13 o The object distance do is positive when the object is in front of the mirror We wonlt discuss the case when the object is behind the mirror 0 The image distance di is positive when the image is in front of the mirror real and negative when it is behind the mirror virtual 0 Focal length f is positive when the focal point is in front of the mirror as with a concave mirror and negative when it is behind the mirror as with a convex mirror With these in mind7 the mirror equations are le Sl K5 3 o E o 8 Refraction and Lenses 81 The Index of Refraction Light will propagate through transparent materials like water or glass much as it does through a vacuum7 but the speed of light is signi cantly less lfthe speed of light through the material is 71 then we de ne the index of refraction for the material as C n 7 71 Since 71 lt 67 we must have n gt 1 For example7 for water7 n 133 82 Snell s Law When a beam of light strikes an interface between two transparent media7 it is bent7 or refracted The angles of incidence and refraction7 both measure rom the normal to the plane surface between the two media are not equal but are realted by 711 sin 91 n2 sin 92 Here7 91 and 92 are the angles in media 1 and 2 and 711 and 712 are the respective indices of refraction of the two media This relation is known as Snell s Law If a ray passes from a material which has a larger 71 to one which has a smaller 71 the ray bends away from the normal Eventually you will reach an angle of incidence within the optically thick77 material for which the refracted ray goes at 90 to the normal It cant go any more than that The angle of incidence here is called the critical angle for the two media and one can show that it is given by the formula 712 sin 0cm 7 711 where the incident and refracted media are 1 and 27 respectively 83 Dispersion of Light It turns out that the index of refraction in most materials is slightly different for different colors of light So when a beam of white light strikes an interface the different colors will bend at slightly different angles We call this phenomenon the dispersion of light We can see this effect with a piece of glass which has non parallel edges The light which emerges has rays for different colors going off at different angles The phenomenon is perhaps most well known from how it causes a rainbow to appear when light from the sun goes into a region of suspended water droplets and then refracts and re ects within the droplet back to the viewers The rays that emerge are dispersed because of the the different refraction angles for the different colors 84 Lenses One can fashion pieces of clear glass or plastic sysmmetric about an axis such that rays which arrive at the device parallel to the axis are all bent to meet at the same point Such a device is a converging lens Similarly7 one can make a device such that rays arriving parallel to the axis seem to be diverging from the same point on the arrival side Such a device is a diverging lens For the devices the distance from the lens from the points of convergence or divergence the focal points is the focal length of the lens In discussing the formation images by lens I will assume that light is coming in from the left it passes thru the lens and then is detected seen by someone on the right side of the lens For a converging lens7 if a ray comes in parallel to the axis7 it exits along a line which passes through the right focal point If it comes in along a line passing through the left focal point it goes out parallel to the axis For a diverging lens if a ray comes parallel to the ais7 it exits along a line passing through the left focal point If the ray comes in along a line which passes through the right focal point7 the ray goes out parallel to the axis For both type of lenses7 ray which passes through the center of the lens keeps going in the same direction We can nd the location of an image by tracing some rays that come a particular point on the object Using the behavior of the lenses we draw the following rays 0 A ray which comes in parallel to the axis goes out along a line which passes through a focal point 0 A ray which comes in on a line through a focal point goes out parallel to the axis 0 A ray which comes in on a line through the center of lens continues through the lens along the same line If these rays meet at a point on the right side than a real image is formed lf when tracee backwards they seem to come from a point on the left side7 then that ctional place of origin is the Virtual image The image may be upright or inverted The magni cation m is the ratio of the the height of the image to that of the object m Iiiho If m is negative then the image is inverted 15 85 The Lens Equation As with mirrors7 we can solve a simple equation to relate the distances to the object and image and the focal length In fact the formula has the same form as the mirror equation7 but we must interpret the symbols correctly Our conventions are 0 The object distance do is positive when the object on the left side of the lens We wonlt discuss the case when the object is on the right 7 o The image distance di is positive when the image is on the right side the opposite side from the object and negative when it is on the left the same side as the object Be careful with this convention it would seem to be the opposite of what we use for a mirmr 0 Focal length f is positive for a converging lens7 and negative for a diverging lens With these in mind7 the lens equations are 86 Fixing Human Vision The human eye contains a lens which can change its focal length so as to focus the images of object on the back of the eye The lens should be deformable enough so that objects very far away can be focused and objects as close to your eye as7 say7 25 cm can be focused But there are many among us whose eye muscles donlt work so well and these limits are not attainable7 at least not without help People who have trouble getting distant objects to focus are nearsighted7 or myopic They can focus objects only out to a particular nite distance this is the distance to what is called the persons far point The persons vision can be corrected with a lens the idea is that this lens takes an object at in nity and makes an image at the persons far point With such a lens the person can focus all closer objects so the vision problem is solved arti cially If the distance to the far point is Dfar then we have do 00 and 031 inaI minus sign because image is on same side as object and then the lens equation gives 1 1 1 00 Dfar f so that f inaI Thus the problem of myopia is solved with a diverging lens People who have trouble seeing close objects are farsighted7 or hyperopic These people would like to focus objects which are as close as7 say7 25 cm but they can only focus on objects as close as a place which called the near point7 which is farther than 25 cm This vision defect can be corrected with a lens which takes an object at 25 cm and makes an image at the near point7 where the eye can deal with it properly Because of the fact that 1f shows up in the lens equations7 opticians prefer to use it as a measure of the focusing power of the lens7 and refer to it as the refractive index It has units of m l7 which they refer to as a diopter So they tell me 16 9 Interference and Diffraction Waves are different from particles in that combining separate waves doesnlt necessarily make a bigger wave We get a wave of greater amplitude if the separate waves are mostly m phase and we get a wave of smaller amplitude if the separate waves are mostly Out Of phase We can demonstrate the wave nature of light by passing a beam through a small opening or a pair of openings close together The beam will spread out from the openings in such a way that some locations will receive a lot of light and some very little If the light goes on to strike a at screen there will bright and dark spots on the screen often called fringes To do these experiments it is necessary to have coherent light This means that the light behaves like a wave with a de nite phase with well de ned maxima and minima in the amlitudes of the elds The natural light around us is at coherent but we can get a coherent beam from it if the light passes through a narrow opening Otherwise such as in our lab exercises on this subject we can use the light from a laser which is coherent 91 The TwoSlit Experiment The clearest demonstration of the wae nature of light is the twoalit experiment where coherent falls on two very narrow openings in a barrier We need to assume that the size of the openings is much smaller than their separation which is also pretty small The light which comes from the two holes goes on the strike a screen where a pattern can be observed The pattern of bright and dark fringes occurs because at some points on the screen there is constructive interferece from the light which originates at the two holes at other places there is destructive interference When the difference in path lengths to the two holes is an integer number of wavelengths then the former occurs and when the difference in path lengths is g g wavelengths then the latter occurs lf 9 is the angle separating a point on the screen from the center of the pattern then we have A Bright fringes sin bright mg m 0 1 2 3 1 Dark fringes sin dak m d m0123 92 Diffraction The Single781it Experiement It is also true that coherent light which passes through a single slit in a barrier will make a brightdark pattern on a distant screen A Dark fringes sin 6dark mi m 1 23 w 10 Modern Physics The laws of physics as written around 1900 were apparently complete and successful These laws included Newtonls law of motion and the laws of electricity which we saw this semester 17

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