23-1 (SJP, Phys 1120)ELECTRIC FIELDS:Electrostatic forces are (like gravity) "Action at a Distance". It's a strangeidea! One way people have developed to help get more comfortable with thisidea is another concept: force fields. (Michael Faraday invented this in the1800'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 chargeq somewhere, a distance r away.Little q is pushed (by Coulomb's law), it feels a forceF_qQ = kq Q/r^2. (Read that as the Force on q by 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 comealong. The field isn't exactly physical; it's not something you can taste, orsmell, or see. It just manifests itself if you put a charge (any charge) qsomewhere (anywhere!) (Thus, Faraday doesn't quite think about electricforces as "action at a distance": it's not so much that Q is pushing on q, faraway, it's more like Q produces an electric force field everywhere, and it's thatfield right wherever q is, that finally pushes on q. ) SoElectric 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!)Suppose you have a bunch of charges. If you bring in onemore little charge, "q", anyplace you like, say the littleblack spot, it will feel an electric force in some direction.(You could figure it out by using Coulomb's law, adding thefour separate forces, as vectors. You'll get SOME answer.)But again, you can instead think of an "electric force field" at the point, whichtells you exactly which way a little "test charge" q will be pushed IF you put itthere. The E-field is present whether or not you bother putting q there. It ispresent at any (and every) point in space.I'm going to first just define the E-field mathematically. (I'll justify andexplain this definition on the next page) It should certainly be a vector (i.e. ithas 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 Q q Q1 Q3 Q2 Q423-2 (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 matterhow much charge I use to test for it. If I bring in "q", I'll feel some force F. IfI bring in "2q", Coulomb's law says I'll feel exactly twice the force, 2F. Butsince E=F/q, in the second case (twice the force, twice the charge) the factorsof 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 notthere's any charge physically at that spot - and you can use it to figure out theforce 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, andpay $4. On day 2 you buy 3 pounds, and pay $6. What you pay depends onhow much you buy. So, it might seem complicated to try to predict how muchyou'll have to pay tomorrow, when you buy yet a different amount. But notice:$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 muchyou buy: Price = ($2/lb)*(amount you buy)It's basically the same with E fields: E is like the "unit price per pound" (onlyhere it's really "unit force per charge") Price per pound is a "universalproperty 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 youput in different q's. But then you notice thatForce=(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 thestore.Bottom line: if you know what E is at any point in space, you can immediatelyfigure out the force on ANY charge "q" placed at that point, becauseF = q E.(Just multiply both sides of the equation defining E by "q" to get this.)23-3 (SJP, Phys 1120)Electric Field LinesSuppose 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 fieldpoints radially outward (radially means directly away fromthe 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)This kind of drawing is a little tedious, and picking points at random doesn'tseem like the best way of drawing an E field. But alas, E is definedeverywhere, and it's a vector, so you really can't draw the field in any easyway!There is a neat pictorial trick that people use to try to "visualize" E-fields. Youdraw "lines of force". This means, instead of drawing vectors at points, youdraw 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 youthe direct of E. (The lines have arrows, to eliminate any ambiguity indirection)The more lines you have (the denser they are), the stronger the E field. (Ifyou double the charge, you double the "density of lines".) (Technically, this isthe 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. Butthere 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 thecharge, which tells you the E field is weaker out there.(this is just Coulomb's law, E drops like 1/r^2!) QEEEE +Q23-4 (SJP, Phys 1120)Example 2: A single - charge.The arrows all point towards the - charge(E fields go towards negative charges)The lines are all radially inward.Notice that the lines are again less dense further away,(this is just Coulomb's law, E drops like 1/r^2!)Example 3: A single charge of -2.Just like the last one, but the density of lines is twice asmuch, because the charge is twice
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