29&30-1 (SJP, phys1120) Electric Potentials: We've been talking about electric forces, and the related quantity E = F/q, the E field, or "force per unit charge". In 1110, after talking about forces, we moved on to work and energy Quick Review of work and energy: The work done by a constant force, F, moving something through a displacement "d", is ! W =r F "r d = F||d = Fd cos# More formally, if F varies as you follow some path: ! W =r F " dr r # . E.g. if you (an "external force") lift a book (at constant speed) up a distance d, Newton II says F_net = ma, i.e. F_ext - F_g = 0 (because, remember, if speed is constant => a=0) or F_ext = mg. You do work W_ext = F_ext*d = +mgd (The + sign is because θ is 0 degrees, your force is UP, and so is the displacement vector) The gravity field does W_field = -F_g*d = -mgd (The minus sign is because θ is 180 degrees, the force of gravity points DOWN while the displacement vector is UP) The NET work (done by all forces) is W_ext+W_field = 0, that's just the work-energy principle, which says W_net = ΔKE (=0, here) You did work. Where did it go? NOT into KE: it got "stored up", it turned into potential energy (PE). In other words, F_ext did work, which went into increased gravitational potential energy. For gravity, we defined this potential energy to be PE = mgy, so !PE = mg(y_final - y_initial) = + mgd (=W_ext) (The change in PE is all we ever cared about in real problems) Summary: If you do work on an object in a "conservative" force field: !PE = W_"by you" = -W_"by the field"= ! "r F field# dr r $ Knight generally uses the symbol "U" for "Potential Energy", by the way. ! d F F_ext F_g = mg29&30-2 (SJP, phys1120) Now, let's drop the book, and see what happens. There is no more "external force" touching the book (like "me" in the previous example), only gravity acts. (Neglect friction) Energy conservation says PEi+ KEi= PEf+ KEf, i.e. mgd + 0 = 0 +12mvf2. This formula gives a quick and easy way to find v_f. The concept of energy, and energy conservation, is very useful! Another way of rewriting that equation is (PEf! PEi) + (KEf! KEi) = 0,i.e. "PE + "KE = 0, or "Etot= 0 If only conservative forces act, !U (i.e. !PE) is independent of the path taken. (End of quick review of work and energy!) There is an electric "analogue" of the above examples: Consider 2 charged parallel metal plates (called a "capacitor"), a fixed distance d apart. Between the plates, E is uniform (constant), and points from the “+” towards the “-“ plate. Imagine a charge +q, initially located near the bottom plate. The force on that charge is F_E=+qE (down, do you see why?). (Let's totally neglect gravity here!) Now LIFT "q" from the bottom to the top, at constant speed: You do work W_ext = ! r F ext"# dr r = Fextd = qEd The Electric field does W_field = ! r F E"# dr r = FE d cos(180) = qEd(-1). (Do you understand those signs? Think about them!) Just like the previous case: you did work, but where did it go? As before, it didn't turn into KE, it turned into potential energy. We say the charge's electrical potential energy has increased: !U= U_final-U_initial = qE(y_final - y_initial) = + qE d (= +W_ext) (where y is the distance above the negative plate) We lifted the charge from a region of LOW PE (near the "-" plate) to a region of HIGH PE (near the "+" plate). (Note: "up" and "down" are irrelevant here, you could turn the picture on its side or even upside down. It's not gravity in this story, it's 100% electrical energy.) F_g = mgifd E +q i f d29&30-3 (SJP, phys1120) Just like we defined E=F/q (dividing out q gives force/unit charge), let's now define something we call "electrical potential" or just "potential" = V = PE/q . • Calling this quantity "potential" is really a VERY bad name, because this "potential" is DIFFERENT from "potential energy". • Potential has units of [energy/charge] = [Joules/Coulomb] = J/C. We call 1 J/C = 1 Volt = 1V (People use the symbol "V" for the unit volt, as well as for the quantity itself, Another bad choice, but we have to live with it) "Voltage" and "potential" are basically synonyms. A change in potential is called a "potential difference", ! "VAB= VB#VA="UABq= #r E $dr r AB% and from this, the change in potential energy !U = !PE = q !V. • Potential is a number, a scalar. You assign this number to places, points. • "Potential difference" is defined even if there are no charges moving around. • The sign of potential differences is meaningful! Example: A car battery maintains 12 V between the terminals. If the headlights contain a 36 W bulb, how much charge is the battery moving through the bulb each second? (And, how many electrons is that?) Answer: 36 W = 36 Watt = 36 J/s. Each second 36 Joules of energy are dissipated in a bulb. This energy all comes from the loss of potential energy as charges flow from one terminal, through the bulb, to the other terminal. If a charge "q" drops 12V, the energy lost is !PE = q!V, or q*12V. Each second, 36 J are lost, i.e. 36 J= q*12 V, or q = (36 J)/(12 V) = (36 J)/(12 J/C) = 3 C. That's a lot of electric charge being moved by a car battery! The number of electrons going through the bulb each second is 3C/(1.6E-19 C/electron) = 2E19 electrons. (A heck of a lot) I was a little sneaky about signs (the charge of an electron is negative): just think about it. Here's a related question for you: given that it's (negative) electrons that flow out of a battery, which way do they go? from the "+" terminal through the bulb to the "-", or the other way? (The answer is from - to +. Electrons are repelled from "-", and attracted towards "+". ) + -29&30-4 (SJP, phys1120) For a parallel plate capacitor, we just found (two pages ago) !PE = + qE d, which means !V = !PE / q = (qEd) / q = E d. Here's another sketch of a capacitor: With gravity, you can choose to call "zero" potential energy wherever you want. You might choose sea level, or the tabletop, or the ground. It's the same story with electricity: you can pick any spot you want and call the electrical potential energy 0 there. We usually call this point "the ground"! Let's call point "a" in the diagram "the ground" or "0 potential". Now put a charge "+q" at the point "b" in that figure. That point has a potential given by V(at point b) = E*d. The charge has a potential energy at point b of PE = +q*V(at b) = …
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