» Syllabus

Topic 2 (Electric Potential Energy, Potential, & Capacitance

Home-Work Set for Topic 2 :

PRACTICE QUESTION SUGGESTIONS ...( NOT for grading ...) from Young 9th edition
format codes :
      bold : Basic Bread-and-butter ... you'd better know how to do `em
      Italic : Important Ideas In here ... at least read `em
      Underline : Understand the Underlying structure ... not just the answer

ch.18 conceptual quests 1,2,3,4,6,7,8,9,10,13,14,15 ;
      multiple-choice 2,3,6,8,9,10,13,14 ;
      problems 1,3,4(watch ±!),6,7,8,9,10,11,12,13,14,15,17,18,20,21,22,24,25,27,29,31,34,35 ;
      . . . 39,40,42,43,45,47,48,49,50,53,58,60,61,63,64,65,67,69,71,73,75,76,77,78,79,84,86,87

Topic 2 First Homework set (look at it seriously before Monday's class!) , due 5pm Jul.16 (Mon) on paper :
1. a 2[μC] point charge "A" is held at the origin (x=0,y=0) while point charge "B" (−3[μC]) is moved from x=0.10[m]
      to x=0.40[m],y=0.30[m].
. . . a) calculate the Electric potential at x=0.10[m] caused by A . . . b) and the V caused by A , at B's ending location .
. . . c) calculate the change in Electric PE . . . . d) did B require Energy to move, or release Energy as it moved?
2. Rutherford shoots a He nucleus (called an "α": 4×6.67E−27[kg], +2[e]) with speed 1E7[m/s] at a gold nucleus (from ∞ far away).
      (gold nucleus : +79[e], 197×6.67E−27[kg]) ... ignore the gold's recoil speed   (⁴/₁₉₇)×1E7[m/s]   (=> 2% of the Energy)
. . . a) how close does it get before it stops? (subatomic realm has no friction ) . . . b) what acceleration does it have there?
. . . c) how much KE does it have when it is at r = 21.9E−15[m] from the gold nucleus?
3. The Electric potential inside a typical animal cell is 0.085 [V] more negative than V outside the cell.
. . . a) How much Work does the cell do (±!) , as it "pumps" a K⁺ ion from inside to outside? (in [J] and in [eV])
. . . b) calculate the Electric field (vector!) within the 10[nm]-thick membrane which separates inside from outside ; notice its units!
. . . c) treat the cell as (roughly) a 5[μm] cube; what Area does its membrane have, and how much charge (±!) is on its inside?

Physics II Topic 2 Summary

Idea #4 - - - Charge Q is Source for Electric Scalar Field (Potential), V - - -

#4.a: Charge carries an Electric Potential - - - abbreviated   V , units   [Volt] - - - - - - - -
Nonzero Q modifies the environment near itself ; Electric Potential surrounds it
      is positive near a positive charge ... negative near a negative charge
. . . this is similar to Gravitation's Potential Well produced by Planets or Stars

This electric Potential V is only the "environment portion" of the Electric Potential Energy (see #5 below)

But we have positive and negative charges (not just positive masses) so the electric Potential can be ± also.

#4.b: V weakens with distance as 1/r - - - - - - - - - - - - - -
      => V = k_C·Q_source/r .
. . . r is the distance from the source charge Q to the place where the V is calculated
      recall that Gravitation's potential also weakens with distance :   Γ = −GM_Earth / r .

Every source charge (nearby) contributes to the Potential field at some location ;
. . . simply add all the contributions, algebraically (keep track of + or − signs)
      of course each charge is divided by its own distance from the place of interest .
. . . charges that are not "nearby" don't contribute to V ... kQ/∞ → 0
      but a charge that is 20× as far away still contributes 5% (=1/20^th) as much potential.
. . . here's a graph of the electric Potential (vertical axis, arbitrary scale)
      due to a positive charge on the left and negative charge on the right (x to right, y perspective into screen)
      you can tell that they're equal amount but opposite charge because their total Potential V = 0 midway between them.

#5: Charge q in an Electric Potential has Potential Energy

#5.a: Electric Potential Influences charge q - - - - - - - - - - - - - -
When any electric charge q is immersed in an environment having Electric Potential,
      that charge has electric Potential Energy   U_E .
      => U_{of q} = q V . . . similar to gravity's  U_{of m} = m (−GM_Earth / r )
. . . this influenced charge {lower case} does NOT contribute to the V at its own location
      (charge cannot give itself Energy ... we cannot divide by zero distance)

Negative Potential Energy occurs when the influenced charge has been attracted to the Source charge for a ways.
. . . unless its Kinetic Energy is large enough (so that its total Energy is positive), it is trapped by the Source
      since they cannot become "very far apart" ... at ∞ distance their U must be 0 , and KE = ½mv² can't be negative.
. . . this should remind you that we have negative gravitational U, so are trapped down here in Earth's Gravitional potential well.

Positive Potential Energy means that the two charges have been repelled from one another as they've been pushed together,
      since they must have the same sign (−q in a negative V region, or +q near a positive Q)
. . . positive charges are always "trying to move toward" lower (−) potential ... to be closer to negative charges
      while negative charges are always "trying to move toward" higher (+) potential ... sort of like bubbles rising
. . . the positive Potential of a pair of Electric charges transforms into their Kinetic Energy as they flee from each other.

- - - Vector and Scalar Fields are Related - - -   −F·Δx = ΔPE - - -

δV Related to the Electric Vector Field - - - - - - - - - - - - - -
The difference in Potential between one location and another , δV , is given by −E·δx .
      => If Electric Potential difference is thought of as "g·h" on a topographic map, then
. . .     E would be the slope pointing in the steepest "downhill" direction , as the gradient of V ... −δV/δx
      OR => we might think of the Potential difference as being the "E-field that would be accumulated"
            along (parallel to) the path between the points
. . .     this follows from the Work that would be done by the E-field on a charge which traversed that path.

Within a conducting material, the (static) Electric Field is zero , so E·dr = 0 .
=> Electric Potential V is a uniform value all through that conductor.
NOTE : "Voltage" means Potential Difference ... NOT the Potential at one location !
. . . "Voltage at a location" must be relative to the "0" Potential at infinity, or somewhere else (in a circuit, especially)

Implication ... C = Q/δV - - - - - - - - - - - - -

conducting geometry has Capacitance to store Charge
Capacitance C = Q/δV ~ κA/4πk_cd
. . . this is well-defined since any conductor has only one value for its potential (at any time)
. . . large surface Area A has large Capacitance property ; a lot of charge can spread out on it
. . . d is the distance along the E-field , a "length scale" of the conductor
      small d implies that δV is small"close to zero" , which makes the capacitance large

Capacitance depends on Geometry and material properties - - - - - - - - - - - - - -
the charges carry the E-field which determines the voltage, so the ratio of them is only size and geometry .
. . . but di-electric material in the gap becomes polarized, which reduces the E-field within it, to 1/κ of the external Field
      => voltage is reduced for the same amount of charge , so capacitance is increased (κ × C_o)
. . . many dielectric materials can withstand more intense E-fields before breakdown ("spark thru material") than air can,
      so the gap can be thinner (which decreases the voltage even more, for the same charge on the same Area).

Because the voltage across a capacitor changes as the charge accumulates on it (or leaves it),
      the PE stored in a capacitor is PE = q_total·δV_average = ½ q_total·δVmaximum .
      ... should remind us of the Potential Energy stored in a spring (½ x_max F_max = ½ k x² )

Capacitors in Parallel : Q's add ... Areas add - - -

This implies ... since   Q = C δV ... and δV's are the same ... (A = C d 4πk / κ) ... that
=> C_parallel = C₁ + C₂ + C₃ + . . .

Capacitors in Series : δV's add ... gap distances add - - -

This implies ... since   δV = Q / C ... and Q's are the same ... (d = 4πk κ A/C) ... that
=> 1/C_series = 1/C₁ + 1/C₂ + 1/C₃ + . . .

Capacitors Store Energy : Σ Q(t) δV(t) Δt   while being charged - - -

This implies ... since   δV = Q / C ... and C is constant while being charged ... that
=> U_{stored in Cap.} = Q_totalV_average = ½ Q_finalV_final = ½ Q_final²/C = ½ C δV_final² .
. . . we should always expect a "½" in any formula that has a squared quantity ... (from averaging, or integration in calculus)

Ignore This Last Section Until Comfortable with the Rest of Topic 2

Idea # 6 : the Electric Field itself implants Energy to the Volume of Space it Occupies

Example: Charged Conducting Shell PE ... 3 ways:

method 1, considering the PE as Work done to assemble the charges onto the shell, from their starting place at ∞
      PE = ½ q V , where V = − ∫ E·dr ... from infinity in to the shell radius R .
      E(caused by Q) = (1/κ) k_c Q / r² (away) , so V = − k_c Q / κ ∫ (−)dr / r ²
=> PE = ½ q (k_c Q / κ R) ... = ½ (k_c/κ) Q ² /R .

method 2, considering the charged shell as a Capacitor :
      PE = ½ Q ² / C ... C = κ ε A / d
      A = 4π R ² ; d = R ... so C = κ ε 4π R => 1/C = k_c/κR
=> PE = ½ Q ² k_c/κR ... (same formula as method 1 )

method 3, considering the PE to be contained in the Electric Vector Field's Energy density :
      PE = ∫ u dV = ∫ u(r) 4π r ² dr , with E = (k_c/κ) Q / r² , for r > R .
      . . . = ∫ (κ/8πk_c) (4π r² dr ) (k_c Q /κ r ² )² ... = ½ k_c Q /κ ∫ dr / r² ... we integrate from R to ∞
      . . . (this Energy density is positive everywhere, so total U must be positive even if we integrate from ∞ inward to R )
=> PE = ½ Q ² k_c/κR ... again ...

This seriously limits how small protons and electrons can be

An electron carries its E-field with it, where-ever it goes ... its E-field Energy acts as if it has inertia ... mass = E/c² .
. . . if all an electron's inertial mass was due to its E-field Energy , mc² = E = ½ k_c e² /R yields a minimum Radius
      for the electron's charge distribution called the "classical electron radius"
=> R_e,E ≥ ½ k_c e² / mc²   =   1.4 femto-meter ... about the size of a proton or neutron .
. . . if its charge is spread within its Volume, there would be E-field Energy inside its radius also
      - so R would need to be 3/2 × larger   (2.1[fm] radius)
. . . if some fraction f of its inertia came from "material mass" (?), the radius would need to be larger : R_e,E /f .
      if some inertia comes from g-field Energy , or from B-field Energy (Unit 2) , the radius must also be larger
=> a "point charge" would have infinite E-field Energy , and would have infinite inertia , so cannot really occur .

yes, the gravitational field contains Energy density in an analogous form
=> Energy/Volume = g·g/8πG   . . . (no negative mass, so gravity has no analogy to "polarizability" κ)
. . . this is not included in Newtonian gravitation (Physics I), but is the key new feature in General Relativity (Unit 4)
      (it becomes important for intense g near neutron stars, and for the immense Volume of the entire Universe)