26-1 (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") 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!) 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. (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 Q q Q1 Q3 Q2 Q426-2 (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.)26-3 (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) 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^2!) QEEEE +Q26-4 (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
View Full Document
Unlocking...