Voltage ( = Electric Potential )


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1 V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is a scala thing, called the o the electic potential. Electic fields and s ae two diffeent ways to descibe the same thing. (Note on teminology: The text book uses the tem "electic potential", but it is easy to confuse this with "potential enegy", which is something diffeent. So I will use the tem "" instead.) Qualitative desciption of The at a point in empty space is a numbe (not a vecto) measued in units called volts (V is the abbeviation fo volts). Nea a positive chage, the is high. a fom a positive chage, the is low. Voltage is a kind of "electical height". Voltage is to chage like height is to mass. It takes a lot of enegy to place a mass at a geat height. Likewise, it takes a lot of enegy to place a positive chage at a place whee thee is high. lowe hee highe hee The electic field is elated to the in this way: Electic field is the ate of change of with distance. Efield is measued in units of N/C, which tun out to be the same as volts pe mete (V/m). Efields points fom high to low. Whee thee is a big E field, the is vaying apidly with distance. high Efield low V Mathematically, we wite this as E = o V = E x x (This equation assumes that the Efield is along the xaxis and that E = constant) The technical definition of involves wok and potential enegy, so we eview these fist.
2 V2 Review of Wok and PE Definition of wok done by a foce: conside an object pulled o pushed by a foce. While the foce is applied, the object moves along some axis (xaxis, say) though a displacement of magnitude. x i θ x f Notice that the diection of displacement is not the same as the diection of the foce, in geneal. Wok done by a foce = W x x = cos θ x = = component of foce along the diection of displacement Wok is not a vecto, but it does have a sign (+) o (). Wok is positive, negative, o zeo, depending on the angle between the foce and the displacement. θ θ < 90, W positive θ = 90, W = 0 θ θ > 90, W negative Definition of Potential Enegy PE: The change in potential enegy PE of a system is equal to the wok done by an extenal agent (assuming no fiction and no change in kinetic enegy) PE = W ext This is best undestood with an example: A book of mass v f = 0 oces on book: case, the wok done by the hand is W ext = foce distance m = +mgh. The change in the potential enegy of the book is v i = 0 m is lifted upwad a height h by an "extenal agent" (a hand which exets a foce extenal to the field). In this h g ext gav = mg PE = W ext = +mgh. The wok done by the extenal agent went into the inceased
3 V3 gavitational potential enegy of the book. (The initial and final velocities ae zeo, so thee was no incease in kinetic enegy.) Quantitative desciption of We define electostatic potential enegy in the same way as we defined gavitational potential enegy, with the elation PE = W ext. Conside two paallel metal plates (a capacito) with equal and opposite chages on the plates which ceate a unifom electic field between the plates. The field will push a chage +q down towad the negative plate with a constant foce of magnitude = q E. (The situation is much like a mass in a gavitational field, but thee is no gavity in this example.) Now imagine gabbing the chage with tweezes (an extenal agent) and lifting the chage +q a distance y against the electic field towad the positive plate. By definition, the change in electostatic potential enegy of the chage is PE = W ext = foce distance PE = + q E y This fomula assumes that the Efield is in the y diection. But don't ty to get the signs fom the equations it's too easy to get confused. Get the sign of PE by asking whethe the wok done by the extenal agent is positive o negative and apply PE = W ext. E +q = q E ( f ) ( i ) distance y hi PE lo PE Now we ae eady fo the definition of diffeence between two points in space. Notice that the incease in PE of the chage q is popotional to q, so the atio PE/q = E y is on q independent of q. Recall that electic field is defined as the foce pe chage : E. q Similaly, we define the diffeence V as the change in PE pe chage:
4 V4 PE of q V, o PE = q V q We showed above that PE = + q E y, so V PE q E y = = = E y. Again, this q q fomula assumes that the Efield is along the y diection, but don't ty to get the signs fom the equations it's too easy to get confused. Instead, just emembe that the Efield always points fom high to low : V = E x (if Efield = constant and is along the xaxis) high Efield low To say that "the at a point in space is V" means this: if a test chage q is placed at that point, the potential enegy of the chage q (which is the same as the wok equied to place the chage thee) is PE = q V. If the chage is moved fom one place to anothe, the change in PE is PE = q V. Only changes in PE and changes in V ae physically meaningful. We ae fee set the zeo of PE and V anywhee we like. enegy joule Units of = [V] = = = volt (V). 1 V = 1 J/C chage coulomb Voltage nea a point chage (This is had to compute, since E constant. Need calculus. See appendix fo math details.) V =? Answe: V( at ) = kq Notice that this fomula gives V = 0 at =. When dealing with point chages, we always set the zeo of at =.
5 V5 What does a gaph of vs. position look like? V V nea (+) chage is lage and positive. Q V nea ( ) chage is lage and negative. If we have seveal chages Q 1, Q 2, Q 3,, the at a point nea the chages is V V tot = V 1 + V 2 + V 3 + (fom PE tot = W tot = W 1 + W 2 + ) Voltages add like numbes, not like vectos. What good is? Much easie to wok with V's (scalas) than with E's (vectos). Easy way to compute PE. Voltage example: Two identical positive chages ae some distance d apat. What is the at point x midway between the chages? What is the Efield midway between the chages? How much wok is equied to place a chage +q at x? V tot = V 1 + V 2 = kq kq 2kQ 2kQ 4kQ Vtot = V1 + V2 = + = + = d/2 d/2 d d d ( ) ( ) The Efield is zeo between the chages (Since E = E + E tot 1 2 V =? E =? = 0. Daw a pictue to see this!) x d
6 V6 The wok equied to bing a test chage +q fom fa away to the point x is positive, since it is had to put a (+) chage nea two othe (+) chages. You have to push to get the +q in place. The wok done is W ext = PE = +q V, whee V = Vfinal Vinitial = V(at x) V(at ) = W ext = 4kqQ d Units of electonvolts (ev) 0 4kQ d The SI units of enegy is the joule (J). 1 joule = 1 newton mete Anothe, nonsi unit of enegy is the electonvolt (ev), often used by chemists. The ev is a vey convenient unit of enegy to use when woking with the enegies of electons o potons. om the elation PE = q V, we see that enegy has the units of chage. If the chage q = 1 e = chage of the electon and V = 1 volt, then PE = q V = 1 e 1V = a unit of enegy called an "ev". Notice that the name "ev" eminds you what the unit is: it's an "e" times a "V" = 1 e 1 volt. How many joules in an ev? 1 ev = 1 e 1V = ( C)(1 V) = J 1 ev = J If q = e (o a multiple of e), it is easie to use units of ev instead of joules when computing (wok done) = (change in PE). Example of use of ev. A poton, stating at est, "falls" fom the positive plate to the negative plate on a capacito. The diffeence between the plates is V = 1000 V. What is the final KE of the poton (just befoe it hits the negative plate)? As the poton falls, it loses PE and gains KE. E q = +e V = 1000 V KE = PE = q V = 1e 1000 V = 1000 ev V = 0 V
7 V7 "Equipotential Lines" = constant lines o a constant Efield, we showed befoe that V = E x, but this is only tue if the Efield is paallel (o antipaallel) to the displacement. If we move in a diection pependicula to the Efield, the does not change: if E, then V = 0. Why? Recall the definition PE W qe q q q q of q ext V = = = = Ex x, whee E = E x is the component of the Efield along the movement, which we call the xdiection. If the signs confuse you, emembe "the hi people look down thei electic field noses at the low people". Anyway V = E. Only the component of the Efield along the displacement involves x nonzeo wok and poduces a change in. Equipotential (constant ) lines ae always at ight angles to the electic field. E equipotential lines
Voltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
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