The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

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1 Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit Joule J) In electicity, it is usually moe convenient to use the Electic Potential V (o Voltage) is electic potential enegy pe unit chage, measued in joules pe coulomb (Unit Volt V) A chaged paticle q exets an electic foce (a consevative foce) in an electic field. If the chage q is not static then wok is done in an electic field by moving the chage q. The wok is popotional to the magnitude of chage (Moe wok is needed to move a bigge chage). This is simila to lift/dop an object against/with gavity. Geneal Physics - PH Winte 6 Bjoen Seipel Definition: The electic potential is electic potential enegy pe chage (like we deived the electic field). V = U/q The change in electic potential enegy pe chage is known as potential diffeence o voltage and is measued in Volt. Only changes of electic potential ae meaningful. V = U/q = -W/q 1V [Volt] =1 Nm/C When we dealing with small chages, thee is anothe unit fo enegy: electon volt = enegy of one electon possesses when it has moved though a potential diffeence of 1 Volt 1 ev = J [Joule] Connection between electic field and electic potential (E=const.) V = U/q = -W/q = q Ed/q = Ed F=q E F=mg U=E el =qed (W=Fs cosθ and U=-W) U=E pot =mgh (when E-field is homogeneous, E is constant vecto eveywhee) Electic Potential Enegy Gavitational Potential Enegy The electic potential enegy incease/decease in the same way as like the gavitational enegy. Only diffeences in potential enegy ae impotant E = V/d= V/ s If E=const. The change in the potential pe distance is constant Example: Paallel Plate Capacito Fo calculating physical quantities it is the diffeence in potential which has significance, not the potential itself. Theefoe, we may choose any abitay point as having zeo potential 1

2 Geneal Physics - PH Winte 6 Bjoen Seipel Enegy consevation When we move a chage in an electic field the total enegy the electic foce is consevative must be conseved. (E kin + E el ) initial = (E kin +E el ) final (½ mv +U) initial =(½ mv +U) final (½ mv +qv) initial =(½ mv +qv) final ½ mv final=(½ mv ) initial + q(v final -V initial ) v final=v initial+ q( V)/m ules: positives chages acceleate in the diection of deceasing electical potential negative chages in the opposite diection In both cases the chage moves to a egion with a lowe potential enegy Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential of Point Chages q q 1 Electic foce F = k 1 1 If we elease test chage it will move (acceleate) away At point B: U A -U B =(qed) A -(qed) B = (Fd) A -(Fd) B = kq q/ A - kq q/ B o V A -V B = (U A -U B )/ q = kq/ A - kq/ B if then V B = Definition: The electic potential fo a point chage kq V = [V ] The electic potential enegy fo a point chage q q U = q V = k [ J ] 3 4

3 Geneal Physics - PH Winte 6 Bjoen Seipel Supeposition of two o moe point chages The potential at a given point in space due to seveal chages can be found by adding the potential fom each chage Geneal Physics - PH Winte 6 Bjoen Seipel Anothe way to daw electic potentials: Contou Maps V A q = k q k q k 3... The dense the lines the moe geat is the change in the potential (Equipotential sufaces, Equipotential lines) The electic field points always in the diection of the deceasing lines No wok is equied to move a chage along an equipotential suface, the electic field at evey point on an equipotential suface is pependicula to the suface. 5 6

4 Geneal Physics - PH Winte 6 Bjoen Seipel The ideal conducto Geneal Physics - PH Winte 6 Bjoen Seipel If both sphees have the same potential Evey point on o within such a conducto is at the same potential. Ideal conductos ae equipotential objects. The electic field is stonge whee the conducto is shaply cuved. Why? Let s assume we have to chaged metal sphees both with the same electic potential. Lage sphee: kσ 4π V = kσ 4π E = Small sphee: = kσ 4π = kσ 4π k?4π 4 V = = k?4π Theefoe:?=σ small sphee = σ kσ 4π E = 4 = kσ 4π 4 highe chage density The potential of a metal sphee is given by kq V = The electic field is given by kq E = Q is the chage distibuted unifomly ove the suface A with a cetain chage density σ Q= σ A = σ 4π 7 8

5 Geneal Physics - PH Winte 6 Bjoen Seipel Capacitos A capacito is a device that stoes enegy associated with a configuation of chages. In geneal, a capacito consists of two conductos, one with a chage +Q and the othe with a chage -Q. Geneal Physics - PH Winte 6 Bjoen Seipel Dielectics A way to incease the capacitance is with an insulting mateial between the plates a dielectic Dielectic slab Capacity is the ability to stoe chage and enegy. The capacitance C is defined as the atio of the magnitude of the chage on eithe conducto to the magnitude of the potential diffeence between the conductos: Q = C V o C = Q/V C: Capacitance [F = Faad] The paallel-plate capacito We can calculated accoding to Gauss law fo a paallel plate capacito the electic field as Q σ A Q E = = = ε ε ε A Qd V = ε A We can calculate the potential as V=Ed Q ε A C = = Qd d ε A Capacitance of a paallel-plate capacito Field lines teminate on negative chage and stat at positive chage on polaized slab. Less field lines ae going though the entie capacito. We educe E-field by a mateial dependent facto the dielectic constant κ E E = Vacuum κ=1; H O= 8.4 κ We know E V V = Ed = d = κ κ V decease with inceasing κ C = Q Q Q = = κ = κ C C incease with inceasing κ V V V κ We can expand the equation fo the paallel capacito: ε A C = κ d 9 1

6 Geneal Physics - PH Winte 6 Bjoen Seipel => Capacitance dependence also on the suface aea A of the plates and the distance d. That how a keyboad woks A dielectic beakdown appeas when the Electic field is too stong. That means stong enough to tea atoms apat. Geneal Physics - PH Winte 6 Bjoen Seipel Electical Enegy Stoage Situation: Paallel plate capacito with a chage Q on its plates and potential diffeence of V. Now we tansfe small packages of chage fom one play to the othe. The change in electic potential enegy would be U = Q V Attention! V is not constant. If we would tansfe anothe chage package we have to calculate with anothe V U= QV av = ½ QV Substance Mica Teflon Pape Pyex glass Neopene ubbe Ai Dielectic stength (V/m) 1 x x x x x x 1 6 With Q = CV U = ½ CV V = Q/C U = ½ Q /C Fo a paallel plate capacito Q = ε E A and V = Ed Theefoe U = ½ Q V = ½ (ε E A) (Ed)= ½ ε E A d A d is the volume between the plates so we can define a density (i.e. a quantity divided by its volume) Electic enegy density U E = electic enegy/ volume = ½ ε E Next time you walk acoss a capet and get a shock fom the dooknob think about the fact that you just have poduced an electic field of oughly 3 million V/m. 11 1

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