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by: Josiane Blick


Josiane Blick
GPA 3.72


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This 33 page Class Notes was uploaded by Josiane Blick on Saturday October 3, 2015. The Class Notes belongs to PHYS 2212 at Armstrong Atlantic State University taught by Staff in Fall. Since its upload, it has received 26 views. For similar materials see /class/217876/phys-2212-armstrong-atlantic-state-university in Physics 2 at Armstrong Atlantic State University.




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Date Created: 10/03/15
PHYS 2212 Notes follow and parts taken from Physics 16 Edition Cntnell amp Johnson and Fundamentals at Physics 17 Edition Halliday ResnickI amp Walker Electricity amp Charge All normal matter is made up of charged particles Charge is just a way to measure how strongly two objects attract or repel one another electrically Charge is similar to mass in that the product of the force between two charges depends on the sizes of those charges and is inversely proportional to the square of the distance separating them just as we saw when we studied the law of gravity Notable differences between the two situations include the possibility of repulsion between like charges all masses attract one another gravitationally 7 even antimatter and the vastly greater strength of the electric force We observe charges to be quantized ie not continuous but having a smallest possible size just as we now know water is not a continuous substance but a large number of molecules in units of the charge on the electron usually represented as e The SI unit of charge is called the Coulomb C we ll de ne it later and it is a very large number of charges In fact each electron has a charge of only l6 x 103919 C That means it would take a few billion billion of them to make one Coulomb By convention the charge on the electron is defined as negative meaning the charge on the proton is positive The signs are opposite but the charges are exactly the same magnitude to the highest accuracy we can measure As always there will be a few enhancements and corrections to the statements above that we ll examine later One is that experiments tell us that we can actually have charges which are 13 and 23 the size of the basic charge 6 when we look at the particles which make up protons and neutrons called quarks These have never been isolated individually but their existence has been demonstrated indirectly For this reason we ll still consider 6 to be the fundamental unit of charge You ve probably known since childhood that like charges repel and unlike charges attract To this idea we ll add the fact that charge is conserved This law is absolute No experiment has ever shown the overall destruction or creation of charge We can separate charges very easily 7 rubbing a glass rod with a piece of fur or your hair with a balloon will demonstrate this These are methods of mechanically transferring electrons from one place to another When you do this the material that gave up the electrons becomes positively charged having lost some amount of negative charge and the material that picks up the electrons is negatively charged The sum of the charges though is still zero assuming the objects were neutral in the first place Conductors amp Insulators Just as we found last semester that certain materials conduct heat better than others certain materials conduct electricity better than others usually we see that good conductors of heat are also good conductors of electricity In a good conductor the outermost electrons of the atoms at least some of them are not tightly bound They are essentially free to roam around the conductor and when they move they carry charge from one place to another generating a current which we ll talk much more about in a few chapters or conduct electricity Among the best conductors are gold silver copper and aluminum they re also very shiny and that s not a coincidence Things like quartz glass rubber most plastics wood etc are not good PHYS 2212 conductors and are therefore called insulators The atoms in these materials hold their electrons very tightly Charging Materials If we put an excess of electrons on an object we say that it has been charged We could also take electrons from the object and put a positive charge on it If we want to try this on a conducting sphere it needs to be insulated from the Earth The reason is that the Earth is a huge conductor and will gladly grab all of the excess charge it can reach This is why we use lightning rods to direct lightning to the Earth 7 it can soak up the charge easily We can charge the object by touching it with a charged rod and letting some of the charge transfer it s easier to say that than to say letting electrons move from the object to the rod if the rod is positively charged or letting electrons move to the object from the rod if the rod is negatively charged The excess charge will quickly very quickly move to the surface of the metal sphere It will do this because that s the way it can get as far away as possible from the other charges of the same sign that came over with it from the rod This is charging by contact We can also charge by induction Imagine connecting the metal sphere to the Earth grounding it and bringing a negatively charged rod nearby The electrons in the sphere near the rod are pushed away and the grounding wire gives them an exit to use Some of them leave the sphere entirely and will only retum when the charged rod leaves What if we break the grounding wire connection before letting the rod leave Now we ve driven electrons into the Earth and they can t get back to the sphere so it is left with an overall negative charge Notice that it s exactly the same size as the opposite charge we ve added to the Earth so we re still not creating or destroying charge just moving it We couldn t do the same trick with an insulating sphere because the electrons near the rod aren t free to move large distances away from their atoms much less leave entirely All they can do is shift position a small amount small on the atomic scale and spend more time on the far side of the atom they re bound to That will tend to make the insulating sphere s surface slightly positive and give a small attraction between it and the rod Coulomb s Law Now that we know there is a force between all charges we should find out the form of it We ll find that it looks almost exactly like Newton s law of gravity except we ll replace mass with charge since it s generating the attractive force As in the case of gravity we ll multiply by the inverse square of the distance between the two things and we ll also need a constant to make the units balance This one is sometimes written as k and its value in the SI system is 899 x 109 N mZC2 notice the similarity in the units between this and G 7 we replace kg by C but that s it Notice that while G is small k is very large This re ects the difference in strengths of the two forces For reasons we ll look at later we sometimes write k in a different form k 14TE 80 where 80 is called the permittivity of free space and equals 885 x 103912 C2 N m2 Therefore we write Coulomb s law as qul 612 2 611612 2 F 2 r 4113801 PHYS 2212 where q and q2 are the sizes of the two charges If they are the same sign we ll get a positive force repulsion If they re opposite we ll get a negative force attraction This force is like all forces a vector That means it has magnitude and direction The direction will be either towards the other charge attraction or away from it repulsion In one sense now that we know the form for the force our work with Coulomb s law is done It s now just one more force we need to consider when determining the motion of an object It s part of Newton s 2nd law as a force to be included in 2 Fx 2 m ax x direction with a similar force in the y and 2 directions We ll nd its components just as we ve done with other forces If we want to know the effect of several charges on another charge we ll use Coulomb s law between the charge we re interested in and each of the others to nd the forces involved then we ll get components add them up etc There s nothing particularly special about this new force For fun let s find the gravitational force between two protons and the electrostatic force between them The charges are both 16 X 103919 C the masses are both 167 X 103927 kg and we ll put them 1039 m apart ie next to each other in the same nucleus We get P ngng 667X10 HfJnfkg2 L67X1047ngL67X1047kg186X1044J Grav 2 r hUUmY while for the electrostatic case we have 230N kq1 q 899gtlt109Nm C216gtlt10 19C16gtlt10 19C 2 FElec r loilsmy Look at the difference The electrostatic force is 1036 times larger That s a trillion trillion trillion times This is roughly the same size difference as the diameter of an atom compared to the diameter of the entire universe If charges move under the in uence of this or any force they will create an electric current Current is just the rate at which the charge in a region changes In an infinitesimally small time we will see that the current I is found from 119 dt The units of current are Coulombssecond or amperes PHYS 2212 Conductors and Shielding If there is a net charge on a conductor too many or too few electrons the charges will all go to the surface in an effort to get as far away from each other as possible since like charges repel and conductors allow charges to move freely What does that mean for the electric eld inside the conductor You can answer this by thinking about it either of two ways rst if there w an electric eld the charges in the conductor would move in response to it The electrons would move in a direction opposite to the electric eld and each one moving that way would act to reduce the electric eld until the eld was zero this happens in a very tiny fraction of a second in a conductor 7 essentially instantly The other way you can see the eld must be zero is to imagine the charges on a conducting sphere They will be spread evenly along the surface with no particular point singled out How would the electric eld know which way to point If there is a cavity inside the conductor there will be no electric eld inside it This provides a way to shield a region from external electric elds This kind of sheltered environment is called a Faraday cage If we put a conductor into an external electric eld it will alter the eld slightly Wherever the eld touches the conductor it will be perpendicular to its surface It has to be 7 a component along the surface would push electrons around in the conductor until they built up and cancelled that part of the eld Everything we ve talked about depends on the free movement of charges 7 if we re in an insulator we can t assume any of this is true The Electric Field Sometimes the concept of a eld is a useful one We ve already used this idea when talking about the weight of an object on Earth s surface What we re really doing is nding the force of gravity between some small mass and the rest of the Earth Since in so many cases we know that the separation between the centers is essentially just the Earth s radius we can do some of the math in advance and multiply the Earth s mass by G and divide by the square of Earth s radius After doing that if we want the weight of an object on Earth s surface we just multiply it by the number we found above which we call g This g is sometimes called the gravitational eld of the Earth All we need to do to turn the eld into a force is to multiply by the mass we re interested in Since we know force is a vector and mass is a scalar the eld must be a vector pointing into the Earth We can use the same idea to nd the electric eld produced by a large charge Q Now it will be force divided by charge rather than mass 5 qo E where go is the magnitude of a small test charge F is the magnitude of the force between Q and go and E is the electric eld produced by Q We ll use the letter E for the electric eld and measure it in Newtons per Coulomb ie if the eld strength at a point is 25 NC and we place a charge of 103915 C at that point the force on it should be 25 X 1039 N Charge is a scalar so this eld is must be a vector We ll de ne it as pointing from positive to negative charges In this picture a proton will look like a pincushion with electric eld lines streaming out of it in all directions Electrons will look the same but the lines will be pointing inward We de ne it this PHYS 2212 way so that the force will equal the charge times E For another electron the charge will be negative and therefore E must be negative also to get a repulsion More lines will mean a stronger eld Electric eld lines gt Isolated positive charge Isolated negative charge The restriction on this idea is that we can detect the magnitude amp direction of the electric eld by measuring the force on a charge as we move it around in the field but we need to use a small charge If we re trying to find the field of a 103917 C charge we won t be able to do that very easily if we re using a 10 C charge to do it The small charge s field will become completely overwhelmed and dominated by the field of the large charge For this reason we will usually talk about the motion of a test charge in the electric field This is a charge which is tiny on the scale of the charge producing the field we want to measure We do this so that we can get a more general result than Coulomb s law alone would give us We avoid many long calculations for the gravitational force by defining g and we ll avoid a similar amount of work by defining E Just as g is independent of the mass we re measuring E is independent of the magnitude or sign of the test charge we use For a point charge then the E field looks like this E q 411380 2 Notice that this field depends on distance this shouldn t be surprising since the force between two charges itself depends on distance Other situations produce other fields What are the rules for drawing field lines First we know they go from positive to negative charges Because of that they can t stop in empty space Also since they represent the path a positive test charge q would follow they can t cross which way would the charge go at the intersection For larger charges we should draw more lines and put them closer together Ifwe put a positive charge near a negative charge they will be connected by many field lines and many others will go out and terminate on or start on far away charges Your book has a picture of this configuration known as a dipole because you have two opposite poles near each other If the two charges in a dipole are the same magnitude and opposite sign the total charge in that region is zero This doesn t mean that the field around the charges is zero PHYS 2212 eyerywhere though The reason rs that aporhtwru be equally far away from both charges oh1y wd w m r 39 dAfferentsxgns th ma t mu attuttt 4 HA Thr uttht the same the charges would cancel and we39dhave ho elou Fro e the drstahee from you to the m her top charge rs shghtly smaller than the dutance from youto the bottom charge The fuel exrsts because the charges doh39t cancel for you In the hrrut oflarge rh rute drstahees the two charges wru seem to be m the same place and wru therefore seem to cancel depehdrhg on your abxlxty to measure aeeurately Nouee that at every porht the e1eethe elde tangent to the eldlmes 39 H quotm quotw th 39 wntteh as pqd calculated from 527 anz PHYS 2212 Electric Fields from Continuous Objects For macroscopic collections of charge we are able to ignore the fact that charge is quantized and treat it as a continuous uid The electric eld produced by this extended array of charges is found by adding up the contribution from each tiny piece of it To do this we need calculus We de ne charge densities in one two and three dimensions A line charge will have a linear charge density Coulombsmeter a sheet of charge will have a surface charge density Cmz and a volume of charge will have a density measured in Cm3 In every case we ll nd the electric eld by doing some form of the integral below E I CE I i2 47 80 r For a line charge we ll let the linear charge density be 7 and write dq 7 ds where ds is an in nitesimal length of the line of charge To make the problem simpler assume the line charge is bent into a circle of radius R and we want to know the electric eld along the axis of symmetry of the circle at a distance 2 above it Pieces of the circle on opposite sides will partially cancel each other s electric elds dE z The only part of the eld that doesn t cancel is the part perpendicular to the plane of the circle make this the z axis To nd that component we have to multiply by the cosine of the angle between the electric eld direction and the z axis This can be written as Z rZ2R2 Performing the integral above which stretches around the circle s perimeter we get z 0056 7 EzjOZ RdEcosO 10an zxds 32 41 8022R2 or writing the total charge on the ring as q we get Ez qz 47 so 22 R232 PHYS 2212 Notice that very far from the ring 2 gtgtR this will look just like the electric eld of a point charge We can turn this circle into a disk carrying a surface charge and see what that does to the electric eld again along the z axis The only big change is that the charge element dq goes from being 7 ds to 039 dA For a circle 61A is just r dr 619 but this problem is symmetric about the z axis so we can do the 9 integration immediately and write dq 039 61A 039 2 T r dr We get ER 2rdr SE 6 Z 7 1 430 0 22 r232 230 22R2 We can nd the eld from an in nite sheet of charge which is sometimes a useful idealization of a real situation by allowing looking at the limit of the above expression as R 9 00 so dE While there s no such thing as an in nite sheet what s really important is that we get approximately the same result when R gtgt 2 which is a possible and not uncommon situation Charges in Electric Fields It s not too surprising that a single point charge should move when it nds itself in an electric eld Our de nition of the eld was just the force experienced by a charge divided by the size of that charge What is also interesting and a little more complex is the behavior of a dipole in an electric eld If the electric eld is uniform the force on each charge in the dipole will be equal and opposite As you ve seen in the rst semester of physics if an object is acted on by two equal and opposite forces its center of mass won t move However if those two forces are not acting on a line which goes through the center of mass they will generate a torque One end of the dipole will move downstream along the eld lines and the other end will move upstream This will cause the dipole to align itself with the electric eld You ve seen basically the same effect if you ve ever brought one bar magnet towards another the magnet you approach will rotate around until its poles are along the same line as the approaching magnet In the uniform eld the force on each charge will have a magnitude of qE and each will generate a torque in the same direction 7 that s why the dipole rotates of qE multiplied by the distance to the center of mass d2 multiplied by the angle between the force amp displacement sin 9 Added together we get PHYS 2212 a d a I qEEsmG 2qus1n6pgtltE It s worth mentioning how important the uniformity of the eld is if the eld seen by the two charges is different the forces won t cancel and the dipole itself will move A point charge has a eld which is muniform everywhere A dipole near an isolated positive charge will rotate so that the negative end of the dipole is closer to the positive charge at that point the forces won t balance and the dipole will move towards the isolated charge Uniform elds are another example of a useful idealization of reality Last semester you probably considered the Earth s gravitational eld to be uniform for the rst few days you studied it To do this you just set the acceleration caused by gravity to be equal to g and assume that it s the same everywhere This is a very powerful approximation since it lets you quickly nd the maximum speed of a falling object or its potential energy or the weight of something on the Earth etc You probably also learned that you can t use g 98 ms2 to calculate the gravitational pull of the Earth on the Moon The reason is that the real formula for the Earth s gravitational eld is When you re looking at things as small as people and skyscrapers the difference between using Earth s radius as r and Earth s radius the object s height as r is negligible As r gets noticeably larger that the Earth s radius the discrepancy grows The fact is Earth s gravitational field is really nonuniform everywhere and the same principle applied to the Moon and Sun gives us the main cause of tides Why is this more of an issue this semester than last It s because of the relative strengths of the electromagnetic amp gravitational forces Only very large collections of mass like the Earth have strong enough gravitational elds to be of interest to us at this level Large collections of mass typically have large on the human scale sizes 7 many km or hundreds of km or more Therefore moving things tens or hundreds of meters humanscale units won t reveal large changes in g The electromagnetic force on the other hand is much stronger and even tiny collections of charge physical sizes much less than 1 meter can produce easily measurable electric elds When we move meters or hundreds of meters from this tiny collection of charge we ve changed the size of the electric eld we measure by a large amount Back to the dipole If it will rotate under the in uence of a eld there must be a potential energy involved The potential energy will be 9 U W Iquot d6 pEcos6 790 PHYS 2212 if we set the zero of potential energy to be the point where the dipole is aligned perpendicular to the eld Gauss Law Finding the electric eld of something other than a point charge can be a dif cult problem Fortunately if we have a situation of high symmetry there is a technique we can use to quickly nd the electric eld s magnitude amp direction To use this technique known as Gauss law we need to introduce the idea of ux This can be thought of as the number of electric eld lines entering or leaving a certain area in space To simplify this idea we ll use an ordinary square region as our area You can see two different situations below in each case the strength of the electric eld density of eld lines is the same and the area of the square is the same The rst diagram shows the square oriented so that the maximum number of eld lines pass through it this makes the ux a maximum In the second picture no eld lines go through the square so the ux is zero Flux is therefore a kind of projection and there will be a Cos 9 term in the formula for it It s not hard to see that a larger square more area would give a larger value of ux and a stronger electric eld more dense eld lines would also give a larger ux The formula is then seen to be IEACOSG where I is the ux E is the electric eld A is the area involved and 9 is the angle between the electric eld and the vector perpendicular to the area More generally we ll want to know the ux through a closed surface this will be the boundary for some volume the siX square faces of a cube form its boundary the 4 T r2 surface area of a sphere forms a boundary for its volume etc If the surface is arbitrary we have to do an integral to nd the ux since A and E won t in general be constant Now we can write 135133722 21 Why do we care about ux The same reason we care about any quantity like this in physics 7 it s conserved or related to something that s conserved In this case the ux is proportional to PHYS 2212 the charge enclosed by our closed surface de ning the region of integration above This is Gauss law and we can write it as 806D80 Ed21Q enclosed This is an incredibly powerful law For example if I want to know the ux through a region of space shaped like a cow all I really have to know is how much charge m of course is contained in that cowshaped volume of space If it s zero I know the ux is zero If I bring a large positive point charge near but not inside that same volume of space lines of the E eld will be entering one side of the cow and exiting the other The density of lines will be greater on the side of the cow nearest the charge Different tiny patches of area on the cow will be pointing in different directions relative to the electric field passing through the cow which will also point in different directions at different parts of the cow This would be a horrible calculus problem and a horrible cowculus problem However if we know there is no charge inside the cow we know that E and each little piece of cow skin dA must be arranged in such a way as to cancel out when the contributions over the whole cow are added In the drawing below if the surface on which we apply Gauss law is the red dashed sphere surrounding the positive charge we can see all sorts of arrows leaving it 7 electric field is owing out of it and there is no balancing electric field owing in If our surface is instead the blue dotted sphere as much electric field is owing out of the right side as is entering the left side The net ux is zero because the enclosed charge is zero If the charge below were negative everything would look exactly the same except for the direction of the arrows which would be reversed A T lt i gt 39 Gauss law H Coulomb s law Both Gauss law amp Coulomb s law talk about the forces or fields which we know are tightly connected produced by and acting on charges To connect them directly imagine that our closed surface in the integral mentioned earlier is the surface of a sphere with a charge q at its center If we give the sphere a radius r and assume for simplicity that q is positive we will have an electric field directed radially outward at all points on the sphere s surface Since the charge is at the center all points on the surface are the same distance r from the charge so the electric field has the same magnitude everywhere on the sphere Also the normal to a tiny piece of the surface 11 PHYS 2212 of the sphere will point radially outward so the cosine term will be one everywhere The only nonconstant thing left to integrate is the area of the sphere so we get 8 0 E 5 61A 2 q The area of a sphere is just 4 7 r2 so we can rewrite the equation above two ways 1 c1280 E4739c r2 0r Ezm Charged Conductors What will happen if we pile charges up on a conductor which is insulated from the Earth so that they can t leave How will they arrange themselves We know that like charges repel each other so they should try to get as far from one another as possible It might seem that the best way to do that is for the charges to distribute themselves uniformly throughout the conductor so that each one has the largest possible volume to itself In fact the charges will all collect on the surface since this will minimize their energy Gauss law can be used to demonstrate that the charges have to appear on the surface The main idea here is that we can t have an electric eld inside a conductor and expect it to last The de ning characteristic of a conductor is that charges are free to move in it If we found a way to apply an electric eld inside the conductor the charges there would just rearrange themselves until their eld cancelled out the applied eld There are situations where electric elds can exist in a conductor but we would then be looking at electrodynamics instead of electrostatics Using this idea imagine that the closed surface in the integral form of Gauss law known as a Gaussian surface is one that is just barely underneath the physical surface of a conductor Whether the shape of the Gaussian surface is the same as the shape of the conductor or some other weird shape that will also fit inside the conductor the E eld has to be zero or the charges will just rearrange themselves until the eld is zero This means that E is zero regardless of our choice of surface so the ux must be zero also If the ux is zero the charge enclosed by the Gaussian surface is also zero The only place left for the charge to be is on the physical surface of the conductor What if the Gaussian surface is larger than the conductor In the case of a sphere the charge will be uniformly distributed on the surface If the surface isn t spherical the charge distribution will be more complicated We can simplify the situation by breaking the surface into many tiny regions each of which will appear to be at just like the surface of the Earth appears to be at on the small scale To make the integral easier our Gaussian surface will be a cylinder with one end inside the conductor and the other end outside The aXis of the cylinder is parallel to the normal of our small piece of the surface The E eld will also have to be normal to the surface because any component of E parallel to the surface would cause charges to move around until PHYS 2212 that component was neutralized So looking at the surface of our cylinder we see that l for the inner face of the cylinder E 0 so there is no contribution to the ux 2 for the length of the cylinder E is perpendicular to that part of the Gaussian surface so there is no contribution to the ux For the final part of the cylinder though we ll have a circular end piece whose normal is either parallel to E if the field lines are leaving the surface the conductor is positively charged or antiparallel to E negatively charged conductor We ll get that qenc 6 A80 80 E where 039 is known as the surface charge density and will be measured in Cmz This will be give us the electric field for any small segment of a conductor although we have to keep in mind that for irregularly shaped conductors 039 itself will vary over the surface of the conductor as charge piles up in some places and avoids others Applications of Gauss Law Gauss law is most useful when we pick a Gaussian surface with the same symmetry as the underlying charge distribution For example if we have a line charge we should make the Gaussian surface a cylinder If this line is infinitely long not as ridiculous as it sounds 7 if your distance to a long thin charged line is small compared to the line s length this is a good approximation the electric field can t have a component along the line s length This means the cylinder s end pieces won t contribute to the ux at all The area of the rest of the cylinder would just be 2 T r h and the cosine term will be one That means Gauss law will look like L 80 E27t rhqm s0 E 27t801quot where M is the linear charge density measured in Cm For an insulating infinite sheet of charge the electric field will again have no component in the plane of the sheet by symmetry which way would it point If the electric field is normal to the surface on both sides we can again use a cylinder as our Gaussian surface with the aXis of the cylinder parallel to the normal and the sheet passing through the middle of the cylinder Now there is no electric field through the main body of the cylinder since the cosine term will be zero here and any contribution will have to come from the circular end pieces each of areaA Gauss law gives 80 EAEAqm s0 E2 280 PHYS 2212 If instead the sheet is conducting we can make a useful device known as a capacitor 7 more about that later by bringing a positively charged sheet near a negatively charged sheet Since the two sheets are conducting the charges on each will be free to move to the side nearest the other oppositely charged and therefore attractive sheet The eld we get in this case is uniform in the limit of infinitely large sheets The fact that the charges are mobile changes things compared to the insulatingsheet case One of these conducting sheets by itself would have its charge spread over both faces to give some net surface charge density 0391 on each side with the same electric field on each side When the two oppositely charged conducting sheets are brought near each other all the charge on each plate moves to the inside surface leaving nothing on the outside This gives a surface charge density twice as large on the inside faces 2 0391 and no surface charge on the outside faces meaning no electric field outside of the plates In effect the charge and electric field that used to be on the outside of a plate gets moved to the inside adding to the charge and field there Finally what about a spherical arrangement of charge The book mentioned earlier that 1 if a charged test particle is outside of a shell of uniform charge the shell acts like a point charge as far as the test particle is concerned and 2 a charged test particle inside a shell of charge will feel no force on it at all You can prove this by imagining a spherical Gaussian surface either inside or outside of a charged shell If the surface is inside the enclosed charge is zero and the electric field is therefore zero meaning no force on a charged test particle If the surface is outside a distance r from the center the electric field will just be the enclosed charge divided by the surface area of the Gaussian sphere This would give an E field of quZC 47580 W which you should recognize as the electric field at a distance r away from a point charge of magnitude qm Potential Energy In the same way that a mass in a gravitational field has potential energy equal to mgh a charge in an electric field has a potential energy The mass would like to move in the direction of the gravitational field and the charge would like to move either in the direction of the eld positive charge or opposite to it negative charge Both gravity and the electric force are conservative remember that that means a decrease in potential energy is matched by an increase in kinetic energy which is why we can define a potential As before work is force multiplied by distance In the case of gravity that gave us mgh near the Earth s surface at least where we can say that the gravitational field is a constant g If the electric field is constant we ll just get the expression below WABZqEab PHYS 2212 where a and b are the initial and nal locations of the charge q As we d expect the work done depends on the size of the charge just as the work done by gravity depends on the mass which is moving from place to place We d like to eliminate the dependence on the test charge We do that by de ning an electric potential which is related to but n0t the same as the electric potential energy All we have to do is divide both sides of the equation above by the charge q This will prove to be so useful that we will give this new quantity work divided by charge a name of its own We ll call it Vfor voltage and the units will be JC or volts Part of the value of this will be that we ll automatically know the change in a charged particle s energy when it goes between two points of different voltage if we know its charge For this reason we ll sometimes refer to voltage as the potential difference We can also write the units for electric field strength in terms of voltage by using 1 NC 1 Vm This is sometimes more convenient for example the potential difference between the two terminals of a car battery is about 14 volts and the terminals are typically around 03 m apart That gives an electric field between the terminals of 14 V 03 m or about 46 Vm An electron moving from the negative pole of a battery AA AAA C or D is accelerated through a potential difference of 15 volts That means that each electron will experience an increase in energy of WqV16gtlt10 19 C15V 24gtlt10 19 J This is a tiny amount of energy of course A more convenient unit of energy when we re actually interested in very tiny things like electrons is the electron volt This is easier because an electron moving through the 15 V potential difference would have a change in energy of 15 electron volts A little bit ofmath will show that 1 electron volt 16 x 103919 J Ifwe want we can find the increase in an electron s velocity when it goes through a given potential by assuming that the potential energy becomes kinetic energy While electrons generally do the moving when there is a potential difference applied to something it was originally believed that positive charges were moving to cause the electric current Electrons are said to have a negative charge though so the ow of electrons is from a negative battery terminal to a positive battery terminal When we look at currents we ll generally refer to them as moving through a circuit from the positive terminal to the negative terminal even though we know that s not exactly what s happening Eguipotential Surfaces and Electric Fields Just as we saw with the gravitational field a charge will only do work or have work done on it when it gets closer to or further away from another charge When we move a mass around on the ground assuming g is constant and the Earth is a perfect sphere we don t do any work against gravity If we move one charge around another charge keeping the distance between the two constant tracing out the surface of a sphere we won t do any work From the formula above for potential every point on the sphere s surface will have the same value of V For that reason we call it an equipotential surface We can move a charge anywhere on an equipotential surface without doing any work You ve seen this idea before 7 on a topographic map contour 15 PHYS 2212 lines connect points of equal height You could move a mass around this line without doing any work Also you see the same thing on weather maps 7 contour lines connect points of equal pressure isobars When lines like this are closely spaced that means there is a more dramatic change in what they measure than in a region where the lines are far from one another When the lines on the weather map are closely spaced near you there will be high winds driven by a dramatic change in pressure over a small distance When there are closely spaced lines on a topographic map you have a very steep incline to go up or down What do we get when lines of equipotential are closely spaced A large electric eld We can de ne the electric eld as the something called the gradient of the electric potential The gradient is a measure of the direction of the most rapid change of something If you re on a hill and set a bowling ball down it will roll in the direction where the height change is most rapid If we let go of a charge at some point it will move in the direction where the electric eld is largest so the potential is changing most rapidly Since negative charges move in the opposite direction that the electric eld points we need to de ne the electric eld like this E2111 ds where dV is the change in potential between two points and ds is the distance between the two points Inverting this allows us to nd the potential from the eld Vf Viz an E The potential difference is always taken between two points but we may set the potential at one point to be zero This is exactly what we do with the gravitational eld if you want to know how fast something is moving just before it hits the ground you need to know its starting point m its endpoint Sometimes it is convenient to make sea level the zero of height but if we were looking at something falling from a desk on the 80 11 oor of a skyscraper to the carpet in the same of ce we d probably measure the height of the desk from the oor rather than sea level No matter how we choose our zero point the difference between initial and nal points is all that matters PHYS 2212 Y L A L urr on the nght perpendlcular For Ml r u see nutn t M l we have at the crosslng7 And vvhat way would a eharge move at that polnt7 Tu wdu V r than surface out the onglnal flel d same helght elf there were a small hlll or valley tn the vvater lt would very qulckly be erasedby the movement ofthe rest ofthe vvater Pntentral Difference created by anmt Charge We shouldn39t be very surpnsedthat a polnt eharge produees lts own eleetrle potenual as vvell as quoteorToth lass nd r also ereate lts own eld D vve h uldr ll L 4 energy 1m th u h th tvvorkls of dlstanEE PHYS 2212 the Earth s radius this approximation will be ne However it is relatively common to work with electric elds which can t be considered constant over the distances that interest us and we can no longer just multiply a single value for the eld by the distance a charge is moved in that eld to nd potential energy As you might guess correctly nding the potential energy in this case will require calculus We need to nd the area under a graph of force vs distance and that will give us work or energy Another similarity between the electric force and gravity is that motion around the charge or mass is free Just as the potential energy doesn t change when we move a mass along the ground at a constant height we won t see the electric potential energy change if we move around a charge without changing our distance to it If we re a distance R away from the point charge the integral we need to do is co Vf V jEdr R L 4780 er It s common to set the potential to zero at in nity so we will be left with V q 4n eor Notice that the potential is a scalar We don t have to worry about components here This can be a real simpli cation when we want to see how a test charge moves under the in uence of lots of other charges We don t have to nd the vector components of the electric eld from each charge and then combine them 7 we can just add the potential difference produced by each one at the location of our test charge to nd the total potential difference The potential produced by a charge will have the same sign as the charge If we have a positive charge and a negative charge near each other there will be a place on the line connecting them where the potential is zero Keep in mind that that doesn t mean a charge placed there would stay at rest 7 the direction of movement depends on the way the potential changes with position and the sign of the test charge This means that the general formula for the potential caused by a group of N point charges can be written as PHYS 2212 Dipoles We ve already looked at the electric eld produced by a dipole 7 what about the potential around one We can use the formula above and write it as Lil 471380 r r Vdipole where r and r just represent the distance between the or 7 charge and the point where we re measuring the potential If the distance to the point of measurement is large compared to the separation d between the two charges the term in parenthesis above is approximately equal to d Cos 9 r2 where the angle 9 is measured from the plane that is normal to the dipole aXis and centered between the two charges This gives pCosG d39 l W 4730 2 A dipole moment p can be induced in a collection of point charges by an applied eld Once that happens the dipole can respond to the eld notice that we might have predicted that an electric eld will have no effect on say a neutral hydrogen atom composed of one negative electron and one positive proton We d have been wrong In fact molecules can induce dipole moments in each other and then respond to the induced moment This is the source of the van der Waals force that you may have heard about in chemistry class Potentials amp Continuous Charge Distributions The general procedure to nd the potential produced by a continuous charge distribution is really just an extension of what you ve seen above We let the sum over all charges become an integral and the size of each charge becomes in nitesimal We can then write V 1 47580 F Your book works out the formula for the potentials due to line charges and charged disks Rather than reproduce that here we just note that we could predict the general form for these things using what we ve already found about their electric elds The electric eld of the line charge was found to be a constant multiplied by Mr where 7 was the linear charge density and r was 19 PHYS 2212 the distance from the line to the point of interest Since potential is basically the integral of the electric eld and we know that the integral of drr is the natural logarithm of r the result found in your book is no surprise Similarly we found earlier that the electric field due to a charged disk was a constant multiplied m z VZ2R2 and if we find the integral of that with respect to z we will get a term proportional to Z Z2 R2 61 as you can see in your book Field from Potential Once the potential is known in an area the electric field is relatively simple to find It points from positive charges towards negative charges higher to lower potential and is perpendicular to the equipotential surfaces There must be some directional dependence since we re finding a vector quantity from a scalar The component of E in a given direction is just the negative of the rate of change of Vin that direction so we can write EFLV E 1 E221 8x y ay 82 Electric Potential Energy A charge in an electric potential has a potential energy just as a mass in a gravitational potential does That potential energy is just the product of the charge and the potential If the potential is due to another charge we can write the potential energy of the system as 9192 4ne0r 20 PHYS 2212 Notice that the sign of the energy works out automatically If the two charges have opposite signs we ll get a negative value of potential energy meaning we have to put energy into the system to separate the charges If the signs are the same we have a positive potential energy meaning repulsion If there are more than two charges the total potential energy is just the sum of the potential energies found using the formula above for each possible distinct pair of charges Capacitors and Dielectrics If we alrange two metal plates so that they are facing each other we can use this setup to separate and store charge We ll have positive charges on one plate and negative on the other This is called a parallel plate capacitor Capacitance is basically the ability to store charge When we pile these opposite charges on the plates an electric field will develop pointing from the positivelycharged plate to the negatively charged one The field lines will be perpendicular to the plates so that our configuration looks like this Using a method we ll examine shortly we can find that the electric field inside the capacitor is given by q 6 8 0 A 80 where q is the total charge on each plate andA is the area of each plate Interestingly the strength of this field doesn t depend on the position between the plates 7 it s no larger close to one plate than in the center In the real world the field will bulge out a little near the edges of the plate In this arrangement each plate will have a charge of the same magnitude but opposite sign The formula we use to determine the amount of charge on each plate is just QCV where Q is the charge per plate Vis the voltage of the battery and C is called the capacitance This factor contains information about the geometry and construction of the capacitor and the 21 PHYS 2212 units of capacitance are called farads 1 F l CoulombVolt Farads are huge units of capacitance so we ll typically use things like microfarads nanofarads and picofarads As C gets larger the amount of charge a battery of a given voltage can separate increases An electric eld will be developed between the plates of the parallelplate capacitor and it is approximately uniform Its magnitude will be E Vd where d is the distance between the plates At the edges of the plates the eld will get a little strange but we ll generally ignore that The electric field will point from the positive plate to the negative one as always The formula for C for parallelplate capacitors derived in your book is just K 30 A d C We ll talk about what K is later If we re in a vacuum it s just 1 Capacitors can also be made in other shapes and the calculus used to find the capacitance of these shapes is shown in your book The general procedure to find it uses Gauss law to find the electric field and the fact that the potential difference is the integral of the electric field over some path You should notice that capacitance calculations always leave you with a term proportional to 80 multiplied by a length Capacitors in Series and Parallel We can combine capacitors in different ways to change the total capacitance of the collection This is very valuable as we can have just a few different capacitors and use them to make almost any value of capacitance we need much like we only have a few kinds of coins and bills in our monetary system but we can combine them to form any amount If the capacitors are all connected to the same two points like across a battery we ll call that a parallel arrangement Take a look at the drawing below for an example Each capacitor is represented by a pair of parallel lines red in this case separated by a small gap This is what an ideal parallel plate capacitor would look like The parallel lines at the bottom blue in this drawing are different in shape 7 one is long and thin the other short and thick This is the symbol for a battery The other lines are just wires black in the picture connecting the circuit elements 22 PHYS 2212 These capacitors are connected in parallel but how are they different from a single capacitor with area equal to the sum of their areas There is no difference For a given voltage V the charge on capacitor 1 is Q CJVand the charge on 2 is Q2 C2V The total charge is therefore Q Q2 CJV C2V C1 C2V The capacitance ofthe two in parallel is then just the sum oftheir individual capacitances Cm C1 C2 If the capacitors are in series interestingly enough we ll see that they combine in a very different way Notice that the right plate of the left capacitor and the left plate of the right capacitor in the drawing below are connected by a wire l l L The plates must have charges which are equal in magnitude but opposite in sign since the arrangement plates of different capacitors connected by a wire would be neutral without a battery That means that the plates facing them also have charges of the same magnitude and opposite sign We ll call it Q and label them all Q Q Q Q l l Since the voltage is split between the capacitors the sum of the voltages across each should be the battery s voltage VVln22L C1 C C Total The capacitance of several capacitors in series is found by the math below 23 PHYS 2212 Since a battery had to do work to move charges from one plate of a capacitor to the other energy is stored when charge is separated The work done to move these charges is not constant but depends on the potential difference between the plates Each electron that s moved changes the potential difference between the plates slightly and more work must be done to move the next one We can nd the total work in terms of the total charge and the average potential difference which is just half of the final potential difference since we started at 0V The total work done and therefore the energy stored is just W Q 12 V but we know that Q CV so we can write U1CV2 2 where U is the energy stored in the capacitor We could find the energy density in the region between the plates by replacing C with the formula for it in terms of its geometry and dielectric constant and replacing Vwith E 61 Making those changes to the formula above will still give us total energy but if we then divide by the volume inside the plates A d we ll get an energy divided by a volume or an energy density 1 KeOA uzggLizz d Volum e E z 1 Ad ZEKEOEZ Dielectrics The purpose of a capacitor is to store charge so that it can be released usually much faster than it was stored later For a large value of capacitance a small potential difference between the plates will allow us to separate a large amount of charge For this reason we would like to find a way to increase the capacitance of a capacitor if possible It turns out that we can do this by filling the space between the two conductors we ll use a parallelplate model again with a nonconducting material called a dielectric what would happen if we used a conductor When a dielectric is placed between the plates the electric field will tend to line up the molecules in the dielectric in such a way as to oppose the capacitor s field Imagine a molecule or atom as a collection of positive and negative charges protons amp neutrons When placed in an electric field the electrons will want to move towards the source of the field since the source is a positive charge and the protons will want to move in the direction of the field since field lines end on negative charges Therefore the capacitor s field tends to stretch the atom very slightly When the positive and negative charges are separated even if only by a tiny amount we get a dipole that has its own field This field will act in a direction opposite to the capacitor s field and the overall effect is to produce a smaller total field between the plates See the figure below for the effect 24 PHYS 2212 Dipoles ovals form in dielectric green in response to eld ofplates E eld of dipoles red opposes eld of plates and makes total internal eld blue smaller Strong E eld between plates The eld between the plates E is reduced compared to its value when the space between the plates is empty E0 by an amount K where K is called the dielectric constant and depends on the composition of the dielectric itself Our formula for K is therefore Reducing the size of the electric eld by placing this material inside the capacitor acts to increase the capacitance of the arrangement In other words the same battery can now separate more charge because of the presence of the dielectric Current When we place a battery in a circuit it will push electrons from its negative end through the circuit doing some work in general and back to the positive end This ow of charges is called the current and is measured in coulombs per second or amperes For example your car battery may be able to deliver several hundred amperes of current to your starter That means that if you picked a point somewhere along the battery cable and could count charges you d see hundreds of coulombs of charge go by every second The formula connecting charge current and time is he dt A battery produces direct current dc which means the current is always in the same direction from the positive pole to the negative pole 7 this is called the conventional current and the idea dates from a time when people thought positive charges were doing the moving Today we know that it s generally the negativelycharged electron which does the traveling and the ow of electrons is therefore opposite to the direction of the current The wall outlets provide alternating current ac in which the electrons are pushed back and forth changing their 25 PHYS 2212 direction many times per second The difficulty or ease with which electrons are set in motion by a potential is described by the resistance of the material containing the electrons Resistance and Resistivity The resistance of a given object depends on a few factors first the composition of the object Wires are made of copper not glass This measurement of the inherent ability of a material to conduct electricity is called resistivity and is usually abbreviated by the letter p For two objects made of the same material but which have different sizes or shapes resistance will also be different Resistance is proportional to the length of the wire or object and inversely proportional to its area In other words a fatter pipe lowers the resistance and a longer pipe increases it The formula for resistance can be written as RZLL A where L is the length of the current path through the object andA is the area of the wire This tells us that the units of resistivity must be ohms per meter From the table in your book you can see that some metals like copper and silver have resistivities lower than 2 x 10398 Qm At the insulating end of the spectrum resistivities are in the neighborhood of 10 2m or higher This is why wire is typically made of copper good conductor covered with plastic very bad conductor If the cord for your lamp was not insulated the two wires would eventually bump into each other and provide a way for the charges to return to the wall without lighting your lamp short circuit If you happened to step on both wires the current would go through your foot The dependence of resistance on area can be seen when we look at the sizes of wire used in various situations The wires inside your radio or computer are typically carrying very small currents and therefore can be very small themselves The main power cable into your house is probably about the size of your wrist since it s carrying more current This is why you sometimes hear that extension cords can be dangerous If you re using a thin cord high gauge you re putting a resistance in between the power and the device You can usually feel the cord getting warm Longer cords should also be fatter to counteract this problem The next time you re in a hardware store or home center ask them for a 50foot extension cord you can use with your stove or clothes dryer 7 these don t exist for this very reason What causes resistance It s collisions between the moving electrons and the rest of the wire including impurities in the wire Imagine rolling bowling balls down hills covered with tree stumps The balls will roll a short distance and then collide with a stump losing speed and possibly going back uphill brie y They ll get downhill eventually but it will depend on the spacing of the tree stumps Current in a wire is also sometimes compared to water in a hose An interesting similarity between the two is that in both cases the speed of the particle water or an electron is very much less than the speed at which the signal travels As soon as you ick a light switch the electricity travels with a speed not far from that of light The individual electrons have speeds because of all the collisions slowing them down that are typically a few centimeters per second or so The speed difference in a hose is also noticeable 7 if the hose is 26 PHYS 2212 lled with water already it will start spraying almost as soon as the faucet is turned on An individual water molecule though takes a while to get from the faucet to the end of the hose The resistance of most substances is temperaturedependent At higher temperatures the resistance increases For some materials they can be cooled to a point where resistance to the ow of current disappears This is called superconductivity and was discovered in the early 1900 s but the highest temperatures at which it could be observed were in the neighborhood of 20 K In the mid 1980 s new kinds of superconductors were discovered which remained superconducting to around 175 K or so This is still very cold but it s much easier to cool something to 175 K than 20 K A material that is superconducting at room temperature would be a major breakthrough and would save an unbelievable amount of energy every year which is now lost to heat Ohm s Law The ow of charge through a circuit is very much like the ow of water through pipes The battery is like a pump that pushes water through one end of the pipes and pulls it out the other end Just as water doesn t ow through the pipe without friction electric charges don t move through ordinary conductors without some loss of energy This loss is called resistance for the obvious reason that the movement of charges is being resisted by the material they re moving through For many materials the resistance of a given size of material is a constant and we get the famous equation shown below VIR This is known as Ohm s Law after the person who discovered it This says that as the potential difference across a piece of material increases the current through that material increases in direct proportion If we connect the two ends of a 15 V battery to a black box we don t know anything about what s inside 7 it just has two wires sticking out of it and get a current of 5 amperes we can then connect it to a 9 V battery and know that we ll get a current of 30 amperes if the box obeys Ohm s law Not everything obeys Ohm s law The units of resistance are volts per ampere V A or ohms The common symbol for the ohm is the Greek capital omega or Q When we re selecting wire for a circuit we generally want it to have the lowest resistance possible and we ll typically assume that the resistance of the wire is zero unless we have a particular reason to worry about these tiny resistances What would our equation above predict for the current produced if we connect the poles of our 9 V battery with a wire of zero resistance The equation would predict an infinite or more accurately undefined current What stops this from happening is the fact that the battery has internal resistance of its own so we don t have to worry about the math failing In reality the internal resistance is usually pretty small so you can get currents that are very large by shorting out the battery Doing this with a car battery is almost guaranteed to injure someone Electric Power We can find the power consumed in a circuit the power delivered by a battery by taking advantage of what we know about energy We know that moving a charge q through a potential 27 PHYS 2212 difference Vrequires an energy equal to qV We also know that energy per time is power We can combine these to get Pz Vle At The unit of power is still what it was before 7 1 watt 1 Jsec or in this case 1 VoltAmpere If the device we re looking at obeys Ohm s law we could also write V2 PIZR Using this what is the resistance of one of the heating elements in an electric water heater if it consumes 2500 W at 220 V It s about 20 9 That means the current must be around 112 A EMF If we want to talk about charges moving rather than sitting still they need a reason to move That reason is called the electromotive force or emf Emf is not really a force though 7 it s just a potential difference between two places voltage which as we know will cause charges to move in response to it The symbol for emf is a script capital E or S If we use a battery to provide the emf it works by means of a chemical reaction that causes electrons to collect at one end and be depleted from the other end The potential difference between the ends or emf is a measure of how badly the electrons want to get back together with the ions If you connect the ends of the battery with a wire so that the electrons have a way to do it many of them will move along the wire to rejoin ions don t do this 7 it will drain your battery and for reasons we ll see later it will also make the wire dangerously hot For our purposes a battery is just a source of emf Circuits Combining resistors with other electrically useful elements will give us an electrical circuit The name is significant because there must be a complete path allowing the electrons to travel from a source of emf a battery for example through the various elements and then back to the other end of the emf source If that path is not complete we have an open circuit and no work will be done When we re designing these circuits we find that it s easier to draw a picture of the circuit than it is to list every element by name Right now our circuits will be pretty simple See the table below for pictures 28 PHYS 2212 Element Symbol Battery I Capacitor 39 Resistor W Straight Wire zero resistance Kirchoft s Rules For complicated circuits which may have more than one battery or generator and multiple current paths we need to use a couple of guiding principles to gure out what s really going on These are called Kirchofl s rules and the reasoning behind them will be easy to see First the junction rule This says that for a given junction place where 2 or more wires connect the incoming currents must exactly match the outgoing currents If this were not true charge would pile up somewhere in the circuit causing an emf which would oppose the driving emf This is really just a statement about the conservation of charge and we can return to our analogy of people in a building You can be sure that the number of people entering a revolving door is equal to the number of people leaving it Mathematically the currents entering a junction will be positive and the currents leaving it will be negative The sum of these positives and negatives is required to be zero Rule 1 The loop rule just tells us that energy is conserved Ifwe follow a charge all the way around a circuit it will see as many potential drops across resistors etc as potential jumps across batteries or generators We ll say that the end of a resistor closest to the positive terminal of a battery or generator is itself positive and the other end is negative If there is more than one source of emf we just guess at the direction of current ow and mark our resistors so that it ows from to When a charge here we re assuming positive charges to make life simpler moves from the positive side of a resistor to the negative side there is a potential drop equal to IR When it moves through the battery from the negative side to the positive side after passing through the rest of the circuit it will experience a jump in potential equal to the battery s voltage The sum of potential drops and potential jumps through the circuit will be zero We have a little freedom with this method since we can pick the direction of the current if there are multiple sources of emf and we don t really know which way it is going in the loop We combine our two rules applying the junction rule at each junction and the loop rule around every loop We ll get multiple equations with multiple unknowns and we then have to solve them If at any point we get a negative value for one of our currents it just means we picked the wrong direction for it 29 PHYS 2212 Internal Resistance As mentioned before a battery or a generator for that matter has some internal resistance of its own This is usually very small but because it will effectively be placed in series with the rest of the circuit there will be a voltage drop across it The true voltage across the terminals is called the terminal voltage and will be lower than it would be without the internal resistance As the resistance of the rest of the circuit decreases the importance of the internal resistance increases Higher current through that internal resistance means a higher voltage drop across it Resistors in Series If we have more than one resistor in a circuit one way we can combine them is to join them at one end If we do this all of the current must pass through each resistor This obviously acts to increase the resistance of the circuit There will be a drop in voltage from one side of a resistor to the other The size of the drop is determined by the current owing through the resistor and its resistance 39 In the diagram above it is clear that the current goes through each resistor In this situation the resistance of the two resistors is equal to the sums of their individual resistances In other words RT R1R2 R3 olal for multiple resistors in series For example if the battery produces a potential difference of 18 V and the resistances of the two resistors are 2 Q and 10 Q what will happen in the circuit The total resistance by our formula above must be 2 Q 10 Q 12 9 That means the current must be VR or 18 V 12 Q or 15 A Think about what would happen between the resistors if the current were different in each For a current of 15 A the voltage drop across the 2 Q resistor will be 2 2 times 15 A or 3 V When we get to the next resistor the voltage drop across it will be 10 2 times 15 A or 15 V Total voltage drop 15 V 3 V 18 V battery voltage so we re OK If you ve seen Christmas lights they were built so that one blown light bulb the filament of which is just a resistor took out the entire string That s because they were wired in series If you break one part of the circuit it s all going to stop Resistors in Parallel If the resistors in a circuit are arranged so that there are multiple separate paths for the current to take they are said to be in parallel PHYS 2212 In the drawing above a given electron will pass through either one resistor or the other but not both This provides a second path for the current to take and therefore reduces the total resistance to its ow If you imagine people leaving a crowded building we can think of them as the charges and resistors as the doors If we have two doors in parallel meaning you can go through either one to get outside more people will be able to leave per second than with only one of the doors open Putting doors in series would correspond to having to use both doors to get out and that will obviously let fewer people leave than if they only had to go through one door or the other The voltage drop across each resistor above is the same as the voltage of the battery they re both connected to the poles of the battery individually if we consider the wires to have zero resistance For different resistances these identical voltage drops mean different currents pass through each resistor If we return to our 18 V battery and resistors of 2 Q and 10 Q we ll find that the current through the 2 Q resistor is 18 V 2 Q or 9 A The current through the 10 Q resistor is 18 V 10 Q or 18 A giving atotal current in the circuit of 108 A If we want to replace this pair with a single resistor which gives the same current ow it would have to be 18 V 108 A or 167 9 We can find that directly by learning how to combine resistors in parallel The equivalent resistance of a number of resistors in parallel is R Total R1 2 3 Let s check this formula on the result we just found We should find that 1 RTom is just the sum of1 2 Q and 1 10 Q or 610 9 Rmal must then be 1 610 2 or 106 2 167 9 This by the way is also the way to determine the mpg a van has to get before it will use less gas on a trip than 2 or more cars 7 remember that for spring break To summarize resistors in series all carry the same current but have voltage drops that depend on their individual resistances Resistors in parallel are all at the same voltage but carry a current that depends on their individual resistances The resistance of resistors in series is always greater than the largest individual resistance The resistance of resistors in parallel is always smaller than the smallest individual resistance Now we see why short circuits are dangerous If the current can go through a wire laid across the battery terminals the resistance in the wire is very low 7 close to zero The circuit has the same resistance it always had The current now has two choices It can take a low resistance path or a high resistance path It will take the low resistance path and lots of current will ow We know 31 PHYS 2212 that the power lost to heat in this wire because it has some resistance even if it is small will be VZR or PR which will be very large when R is small How are the outlets in your house wired They must be in parallel If they were in series all of your lights and appliances would go off when a light bulb blew This also explains why plugging too many appliances into a circuit will cause your circuit breaker to trip Each device you add in parallel lowers the total resistance of the circuit and therefore increases the current When the current reaches a certain value the circuit breaker will trip and the power will be cut off This used to be done by fuses and is still done that way in most cars The idea behind the fuse is to isolate a small piece of wire made of a material that melts at a low temperature If that little isolated piece of wire burns through the circuit is open It s much better for that little piece of safelycontained wire to melt than for a random part of the wire inside your wall to do it Many of the resistive circuits we ll eventually work with can be reduced to collections of resistors in series and parallel Whenever you re trying to decide whether two resistors are in series or parallel just look at the current path if the current has to go through both they re in series If it s required to pick a path they re in parallel Current and Voltage Measurements How can we check ourselves and see if the numbers we calculate for things like currents and voltage drops are accurate As we ll see later a current both responds to and creates its own magnetic field By arranging a coil of wire between magnets we can watch it move as a current passes through it If we put a spring on it to resist its movement and a needle on the coil to track the movement we ve made a device called a galvanometer which is the main part of a current measuring instrument called an ammeter The other important part is a shunt resistor which gives the current an alternate path through the device and thereby diverts large currents most of them at least around the sensitive galvanometer Ifthe galvanometer needle de ects enough to measure small currents we can see that very large currents would cause the needle to go around many times or more likely would burn out the thin wires inside the galvanometer As long as we know the relative resistances of the galvanometer and the shunt resistor we can figure out the total current The ammeter has to be inside the circuit to measure the current since it can only measure what ows through it That means it will be in series with the part of the circuit it s trying to measure That also means that its presence will increase the total resistance and therefore lower the current The process of measurement alters the true value we re trying to measure This is not really unfamiliar to us though If we want to know the temperature of a small cup of coffee and we stick a large roomtemperature thermometer in it we know from last semester s study of heat that the coffee will cool a little as it warms the thermometer up We minimize this effect in the specific heat lab by using quantities of water that are large compared to the mass of the thermometer We can minimize the effect of our ammeter by making its internal resistance as small as possible If we connect the galvanometer in parallel with part of the circuit we will be providing an alternate path for the current If the galvanometer has a tiny resistance we ll siphon off most of 32 PHYS 2212 the current and burn out our ammeter To get around this we put a large resistor in series with the galvanometer to make the current path less attractive to the charges in the main circuit The current that does pass through the galvanometer de ecting the needle can be multiplied by the total resistance of the galvanometer large series resistor to get the voltage between two points Now we ve made a voltmeter Because the addition of a voltmeter gives a new current path it will alter the behavior of the circuit unless the resistance in the new path is very large Therefore we d like a voltmeter to have a very high resistance just like we want our ammeter to have very low resistance Remember that ammeters are wired in series with the circuit and voltmeters are wired in parallel with it The ammeter can be thought of as replacing a tiny segment of the wire in the circuit The voltmeter is connected between two different points to measure the potential difference If we do this backwards we ll have a problem A voltmeter in series with a circuit will make the total resistance huge and will probably stop the current An ammeter in parallel with a circuit will because of its low resistance suddenly find itself routing a huge current between two points For this reason most ammeters have fuses that are designed to burn out first before the galvanometer can be destroyed RC Circuits We ve glossed over this before but when we connect a capacitor to a circuit the charge that it separates doesn t collect instantly As charges ow from one end to the other through the battery they collect on the plates and begin to establish an electric field that opposes the accumulation of more charges This shouldn t be surprising 7 we know that the negative charges on one plate would like to be as far from each other as possible The charge on the capacitor builds up to its theoretical maximum value of Q0 C Vas time passes according to the formula QrQo I em In other words the rate at which the capacitor charges depends on the resistance and capacitance of the circuit How long would it take to fully charge According to the formula and assuming charge doesn t come in fixedsize units of electrons an infinite amount of time As a practical matter we ll see that Q is close enough to Q0 after a few time constants have passed The time constant for the circuit is found by just multiplying resistance and capacitance TRC After one time constant so I 17 the value of the charge will be Q0 1 7 6391 or about 063 Q0 When I 5 E we ll get Q0 1 7 equot or about 0993 Q0 The capacitor will discharge exponentially as well Assuming it starts off charged to Q0 it will discharge according to


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