23 1 SJP Phys 1120 ELECTRIC FIELDS Electrostatic forces are like gravity Action at a Distance It s a strange idea One way people have developed to help get more comfortable with this idea is another concept force fields Michael Faraday invented this in the 1800 s It s strange too but very useful and very powerful A charge Q produces forces on ANY other charges anywhere in the universe Imagine putting a tiny charge q somewhere a distance r away Little q is pushed by Coulomb s law it feels a force F qQ kq Q r 2 Read that as the Force on q by Q q Q Faraday argued there s an electric force field surrounding Q It s like a state of readiness to push any other charge that happens to come along The field isn t exactly physical it s not something you can taste or smell or see It just manifests itself if you put a charge any charge q somewhere anywhere Thus Faraday doesn t quite think about electric forces as action at a distance it s not so much that Q is pushing on q far away it s more like Q produces an electric force field everywhere and it s that field right wherever q is that finally pushes on q So Electric Fields are vectors they have magnitude and direction Electric Fields surround electric charges Electric Fields exist in empty space think of fields as a property of space Q2 Q1 Q3 Suppose you have a bunch of charges If you bring in one more little charge q anyplace you like say the little black spot it will feel an electric force in some direction Q4 You could figure it out by using Coulomb s law adding the four separate forces as vectors You ll get SOME answer But again you can instead think of an electric force field at the point which tells you exactly which way a little test charge q will be pushed IF you put it there The E field is present whether or not you bother putting q there It is present at any and every point in space I m going to first just define the E field mathematically I ll justify and explain this definition on the next page It should certainly be a vector i e it has a size and a direction It should tell you the force on ANY test charge q We define E F q or more carefully E at point p F on test charge q at point p q From this def the units of E will be Newtons Coulomb or N C 23 2 SJP Phys 1120 I did NOT define E F instead I divided out the test charge q Why Because the E field is a property of space at the point p It shouldn t matter how much charge I use to test for it If I bring in q I ll feel some force F If I bring in 2q Coulomb s law says I ll feel exactly twice the force 2F But since E F q in the second case twice the force twice the charge the factors of two cancel and E comes out the exact same no matter WHAT q is That s what we want E has some value at every point in space whether or not there s any charge physically at that spot and you can use it to figure out the force on any test charge of any size that you bring to that spot Analog Interlude to help motivate E fields Go to King Sooper s and buy some sugar On day 1 you buy 2 pounds and pay 4 On day 2 you buy 3 pounds and pay 6 What you pay depends on how much you buy So it might seem complicated to try to predict how much you ll have to pay tomorrow when you buy yet a different amount But notice spent amount bought 4 2 lbs 6 3 lbs 2 lb There is a simple underlying universal common UNIT PRICE So now you immediately know how much you ll pay no matter how much you buy Price 2 lb amount you buy It s basically the same with E fields E is like the unit price per pound only here it s really unit force per charge Price per pound is a universal property of sugar no matter how much you buy even if you buy none The force price on a test charge q seems complicated at first different if you put in different q s But then you notice that Force unit price amount E q Knowing E you can easily figure out the force on ANY q now That s one reason why E is useful it s like knowing the unit price at the store Bottom line if you know what E is at any point in space you can immediately figure out the force on ANY charge q placed at that point because F q E Just multiply both sides of the equation defining E by q to get this 23 3 SJP Phys 1120 Electric Field Lines Suppose we put a positive charge at the origin How might we represent draw the E field We might pick a few randomly chosen points the E field points radially outward radially means directly away from the Q like a radius of a circle The farther away you get the weaker it is because force and thus E drops off like 1 r 2 E Q E E This kind of drawing is a little tedious and picking points at random doesn t seem like the best way of drawing an E field But alas E is defined everywhere and it s a vector so you really can t draw the field in any easy way There is a neat pictorial trick that people use to try to visualize E fields You draw lines of force This means instead of drawing vectors at points you draw lines called field lines Field lines start and end at charges always The direction of the lines really the tangent to the lines at any point tells you the direct of E The lines have arrows to eliminate any ambiguity in direction The more lines you have the denser they are the stronger the E field If you double the charge you double the density of lines Technically this is the number of lines per unit area perpendicular to the field Lines never cross if they did E wouldn t be defined at the crossing point But there must always be a unique force on any test charge Here are several examples Example 1 A single charge The location of the arrows is not significant The arrows all point away from the charge E fields go AWAY from positive charges The lines are all radially outward Notice that the lines are less dense further away from the charge which tells you the E field is weaker out there this is just Coulomb s law E drops like 1 r …
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