»  Syllabus
» 203§2 web page

- - - Unit 1 - - -
» topic 1 (ch.20)
 . . . here! ⇒
» topic 2 (21+23)
» topic 3 (22+23)
- - - Unit 2 - - -
» topic 4 (24)
» topic 5 (½25+26)
- - - Unit 3 - - -
» topic 6 (25+½18)
» topic 7 (18+19)
» topic 8 (17)
- - - Unit 4 - - -
» topic 9 (29+28)
» topic10 (30)

-- MU Physics 2 --
» MU-online
 (BlackBoard site)

» 204 §2 web page
» 204 §2 syllabus

- - Off-Campus - -

» PhysicsForums
(human hw help)

» hyperphysics
 (detailed e-book)

»  Knight 4th ed.
(pdf download)

Topic 1 − Electric Charge property; Electric Force & E-field Readings for Topic 1 (Knight Jones Field 4th ed.): ch.20

Topic 1 Quiz - now to occur Tue.Jan.23 - its header will look like this:

Here are my solutions to Topic 1 Web-work as a pdf

Physics II ForeWarnings Overview

Physics I spends most effort on how objects respond to situations that are presented to them.
      typical example: a 2 kg cart's acceleration, while pulled up a 20° ramp by a 3N Force
. . . In contrast, Physics II spends most effort on how subjects cause such a situation.
      Phy.I example: a 92 N/m spring is stretched 25 mm up-ramp ... what is the 2 kg cart's acceleration?

Physics I objects & subjects are visible items, usually on a human-like macroscopic scale.
      typical exception: 2E27 kg Jupiter in orbit 520E9 m from the 2E30 kg Sun ... what is its orbit time?
. . . In contrast, Physics II causes are invisibly microscopic, and NOT intrinsic to any macroscopic item that is asked about.
      so drawings in Phy.II will be more important for understanding than drawings in Chem.
      most Chem drawings are intrinsic to the name ... we must explicitly label our Phys.II drawings!

Physics I only deals with ONE field (gravity g), usually constant & uniform ...
      ... you might've treated it as a given object property (a specific Force, or sort-of-like a "potential" acceleration).
      Knight's book derived the orbit time formula almost that way; the special-case formula doesn't even have a "g" in it!
. . . Physics II deals with TWO new fields, seldom uniform & often changing ... FEW special-case formulas are worth deriving.
      so most equations on the Phy.II sheet DO have either the vector field , or its scalar potential field.
. . . the formulas are to get a Force or Energy value ... you are supposed to know what to do with that value!

Physics I numerical values are usually 2 or 3 digits from the decimal point - scientific notation and/or metric prefixes were seldom needed.
      most-often-used value was 9.8 Newton/kilogram ... metric units were chosen so the base would be human-scale
. . . the value most-often used in Physics II will be 1.6E−19 Coulomb/e to convert from microscopic to large-scale.
      but a typical lab scale is about half-way (geometrically) between, roughly a nanoCoulomb (1E−9 C)
. . . so we will use prefixes [down: m = milli, μ = micro, n = nano, p = pico, f = femto ... up: k = kilo, M = Mega, G = Giga, T = Tera]
      that you can substitute (or not) into your calculator as [ m= E−3, μ= E−6, n= E−9, p= E−12, f= E−15 ... k= E3, M= E6, G= E9, T= E12 ]
      ... (I intend to never use deci or centi, or deka or hecto: they ruin the big pattern, to only move the decimal 1 place.)
      ... (but some questions on the common final exam might include those depracated prefixes, from other faculty; sorry for that.)

Physics II Topic 1 Summary

Big Idea #1 : Electric Charge - - - abbreviated   Q   or   q   ... for "quantity"- - - - - -

#1.a: object property; macroscopically manipulable, but microscopically intrinsic - - - - - - - - - - - - - -
. . . each proton has positive charge ; Q_proton = e , the "elementary charge unit"
. . . each electron has negative charge ; Q_electron = −e ; same magnitude as proton , but negative
. . . each neutron has zero total charge ... (positive inner core is cancelled by negative outer shell)

1.b: additive quantity - - - - - - - - - - - - - -
. . . but make sure you keep track of positive and negative signs ...
each neutral atom has zero total charge :
. . . +Ze in nucleus ("atomic number" Z = proton number : 1 for Hydrogen, 6 for Carbon, 8 for Oxygen, etc.)
      + (−Ze) in its electron clouds ("shells") surrounding the nucleus
. . . if an atom gains an extra electron, it has become an anion ... now has Z+1 electrons => Q_total = −e  
      ... Cl⁻¹ example : (+)17e + (−)18e = −1e   total charge
. . . if an atom loses an electron, it has become a cation ... now has only Z−1 electrons => Q_total = +e   ... example: Na⁺
      . . . if it has been "doubly ionized" to lose 2 electrons, it now has total charge Q = +2e  ... example: Ca⁺⁺

1.c: conserved quantity - - - - - - - - - - - - - -
. . . the electrons and protons are essentially indestructable (conserved)
      electrons are easy to remove ... from atom's outside edge, low-mass (=> quick a), "small" viscosity
      the "missing" electron carries its charge (thru atom boundary) to where-ever it is now ("electron current")
. . . can be moved around, but the total amount is always the same (zero)
      static electricity charge is easily transferred as one material surface rubs another (different) material's area
      . . . google "triboelectric" (rule-of-thumb: soft polymers tend to collect electrons from harder silicates and proteins)
      in clouds, positive charge on colder ice crystals transfers to warmer snowflakes (depends on Temperature!)
. . . where the charges are ... Σ Q x ... becomes key description for materials
      called "electric dipole moment" (similar to 1st mass moment from Physics 1, whose rate-of-change was momentum)
      polar molecules (water!) ... if aligned and solidified (wax), it is called an "electret" dipole (sort of magnet-like)

1.d: quantized quantity - - - unit   [elementary charge] , abbeviated   [e] - - - - -
      each ion has an integer number of excess electrons (neg.ions) or deficit electrons (pos.ions)
=> so every isolated object (ion, or atom, or lab-sized) has charge Q = N e , where N is an integer (± or 0) ;
      you cannot "split" an electron ; protons can be "split" but the pieces always have ±1 elementary charge, or zero .
      charge is intrinsic to microscopic objects ... look up their charge (and mass, and spin!) in a table of particle properties
. . . but we usually treat electric charge as if it is not an intrinsic property of macroscopic objects . . . they are usually neutral
      and they still look the same (and have pretty much the same mass) after losing or gaining a billion electrons !
      . . . nucleus mass is ≥99.973% of the neutral atom's mass (except Hydrogen, "only" 99.946%)

Macroscopic Unit of electric charge (S.I. , to use in the Laboratory) is the "Coulomb" . . . 1e = 1.6E-19[C]
. . . 1[C] = 6.25E18 e . . . a mole of protons would carry ≈ 96,500 [C] !   ... you can never isolate that much charge.
      tiniest manipulated charge quantities are   1 femto-Coulomb (1E-15 [C]) , with only 6,250 e accumulated ;
      . . . small lab-size charge quantities are a few   pico-Coulomb (1E-12 [C]) ; big lab equipment collects   nano-Coulomb (1E-9 [C])
      . . . a micro-Coulomb (μC = 1E-6 [C]) is a LOT of charge to isolate

Big Idea #2: Charge Q is the Source of an Electric Vector Field ... Q is the active subject's charge quantity

Whenever a negative charge is pulled away from a positive charge,
      a strong Tension permeates the "empty space" between them ; that Tension tries to pull them back together.
. . . this Tension is called the Electric (Vector) Field

This Electric Vector Field ... E-field , for short ... is similar to the gravity field   g   from Physics 1.
      but E is caused by the subject's electric charge property , rather than its mass property.
Electric field and gravity field do not interfere with each other; both occupy the same region (Volume) , each pointing in its own direction.
. . . at any instant in time, each place has one value for the gravity field intensity   g
      examples: −9.8 N/kg ^r (Earth surface) , +1.7 N/kg ^r (Moon surface), 0 N/kg (from Earth, 89% of the way to Moon)
. . . and that same place will have one value for the Electric field intensity   E
      examples: −50 N/C ^r (Earth humid air) , −250 N/C ^r (Earth summer dry air) , +150,000 N/C ^x (inside cumulus cloud) , 0 N/C (inside metal material)
=> because charges move around easily, the E-field can change drastically in a very short time (lightning! in cloud just after example above)

Charge Exudes ("carries") Electric Field - - - - - - - - - - - - - -
Nonzero Q changes the environment nearby ; an Electric Vector Field surrounds it
      E points away from positive source charges ... points toward negative source charges
      strength of E is weaker at larger distances from Q_source ... as E-field spreads out
=> always draw the E vector(s) on your diagram, labeled by the subject item whose charge carries it

    #2.a: Electric Vector Field Weakens as it Spreads from Source

Coulomb's Law - - - - - - - - - - - - - -
      => E = k_C·Q_source/r² (r̂ ) . . . similar to gravity being caused by   g = G·M_source/r² (toward)
. . . r is the distance from the source charge Q to the place where the E-field is calculated
. . . k_C is "Coulomb's constant" :   8.99E9 [Nm²/C²]   =   1.438 [V·nm/e]

we probably ought to write this as Q causes E :   k Q /r² => E

Every source charge (nearby) contributes to the field at some location ;
. . . "simply" add all the contributions as vector arrows (tail-to-tip). (sarcasm here ;-)
. . . a spherical shell of charge can be treated as if its total charge was concentrated at the shell center
      but the source charge shells only contribute to E-fields outside that shell (inside the shell, the opposite sides' contributions cancel).
This is analogous to finding out which pulls the New Moon harder , Earth (from outside) or Sun (from inside) :
      Earth's contribution to g is opposite the Sun's contribution , so their vectors add in opposite directions => the numbers subtract.

Continuous Field Lines - - - - - - - - - - - - - -
For visualization, we may instead represent the Electric Field as lines which diverge from positive Q (use # lines proportional to Q)
. . . where E is weak , drawn lines (spread over large Area) are far apart :
      the intensity (strength) of E-field is proportional to "density of lines" , or # lines piercing a "unit surface"
. . . the lines continue until they converge on a negative charge , where they end (# converging is proportional to −Q) ;
If the total Q showing (on the page) is NOT zero , assume −Q_total is "spread uniformly" at infinity (off-screen).
. . . so that the E-field lines that don't end on the diagram will be spread uniformly at the edge of the diagram :
      seen from far away, E lines should look like they simply diverge outward from (or converge in to) the total charge ΣQ .

This shows that the electric Field diverges from positive charges and converges to negative charges
      E does not circulate anywhere (at least, if the charges are not moving)
=> so we will be able to come up with an electric Potential Energy function for stationary charges in an environment (in Topic 2)

Big Idea #3: Object Charge q is Forced by Electric Vector Field ... q is the passive object's charge quantity

Electric Field Influences charge q - - - E-field units are   [N/C] - - - - - - - -
. . . When any electric charge q is immersed in an Electric Field (a "tense" environment), it gets pushed
=> F_{to q} = q E . . . (similar to gravity's F_{to m} = m g )
. . . this influenced charge {lower case} does NOT contribute to the E-field at its own location
      (charges cannot push themselves ... we cannot divide by zero distance)
      the only way to be sure that there's an E-field somewhere , is to stick a "test" charge there , and see if it gets pushed (what direction?).
      the best way to find out what charge q is , is to stick it in an E-field to see if it gets pushed
... the Electric Field vector is an environment property ; charge q is an object property ; Force is the effect .
    ( => maybe we should write this as   q E => F_{to q}   charge in Electric field experiences Force )
. . . this is a regular mechanical Force from Physics 1, that tends to accelerate object mass - or gets cancelled by other Forces
      just a new way to make a Force - add it in Newton's 2nd Law sum, with all the others!
      ... ("inertia" for the charge property is a bit complicated - wait till Topic 5 Magnetic Induction .)

#3.b. Electric conduction - - - - - - - - - - - - - -
Electric Force tends to cause acceleration of the object which carries q .

. . . conductive material ("metal")allows charge (electrons) to travel easily thru its bulk ... from edge to edge
      typically one conduction electron from each metal atom ... usually uniformly spread (they repel each other)
suppose metal is immersed in an external E-field (say, pointing +^x, rightward)
      (−) electrons are pulled into the Electric Field (leftward, F = −e E) , accelerating quickly to the left
      as they move   (v = a t , they leave (+) ion cores behind (on the right edge)
      charge separation in the metal accumulates until their (−)←(+) E-field contribution cancels the original external field.
  => e⁻ re-arrangement reduces the interior E-field to zero
      ... they stop accumulating as E becomes 0 ... so little inertia they don't overshoot much

. . . insulative material does not ... each electron is held tightly to one molecule (glass)
      but dielectric material allows its e⁻ clouds to displace slightly (into E-field) from its positive ion core
      => induced polarization reduces the interior E-field by a large factor κ (typically < 8 , in most natural solids)
  => so the E-field within the material will be only 1/κ× as intense as it would have been if the molecules didn't polarize.

#1.e: Charge Multipoles - - - - - - - - - - - - describe a charge configuration "statistically" as a distribution

zero-eth moment (monopole) : the sum of all the individual charges ... weighted by x⁰ (=1)
  => Σ Q_i = Q_total , total charge
      even in interesting electrical situations, the monopole moment (the "net" charge) is often zero ...

first moment (dipole) : multiply each charge by its location vector, then add those products
  => Σ ( Q_i x_i ) = Q_total X_avg , analogous to mass moment from Physics 1
. . . if Q_total is not zero, then X_avg is the "center-of-charge"   (analogous to "center-of-mass" from Physics I)
      a meter-stick with 6[pC] at 0.3[m] and 4[pC] at 0.7[m] has Q_total X_avg = 1.8 [pCm] + 2.8 [pCm] = 4.6 [pCm]
      . . . since 10[pC] X_avg = 4.6 [pCm] ,   the center-of-charge is at   X_avg = 0.46 [m] . . . (closer to the big charge)
. . . if Q_total is zero, then QX is called the dipole moment , standard symbol p_e2 , SI unit [Cm]   (not momentum!)
      ( + − ) dipole moment points in negative x direction ... a + charge is at negative x, plus a − charge at positive x . . . . ( + )
      a dipole moment vector points from the negative "center" , toward the positve "center" ... this one points up the page ( − )
. . . the Electric field caused by a dipole is strong in-between the charges ; is complicated near the charges ;
      and at large distances will weaken as 1/r³ . . . (not 1/r²)
. . . example: NaCl ... the molecule is 564 pm long, Na center is 282 pm from the Cl center.
      in a single molecule (gas phase) the Sodium's outer electron is centered 59 pm from the Cl center, 79% of the Na-to-Cl bond length.
      so if we call the Na⁺¹ location as x=0, the dipole moment is (+1e)(0) + (−1e)(282−59)pm + (0e)(282pm) = 223 e·pm = 35.7E−30 C·m.
    water example:   a water molecule with −1[e] at x=0 , and +1[e] at x=+39[pm] , has zero monopole since the charges cancel
      . . . but its dipole moment is (−1e)(0) + (1e)(39[pm]) = 39 [e·pm] (+x direction) . . . quite strong dipole
      . . . that is why water is such an aggressive solvent . . . in SI, p is 6.24E-18 [C pm] , or 6.24E-30 [C m]
. . . an electric dipole moment is torqued to align with an external E-field , with torque   τ = r×F = p_e2×E .
      if they successfully align, the Electric Field caused by the dipole itself always reduces the average E-field locally, to be weaker than it was before.
. . . atoms and molecules that are polarized by an external E-field ... that is called "induced" polarization ... are already (automatically) aligned that way.
      some molecules are easy to polarize - they are said to have a high "ability" to become polar - but it is an external E-field that actually made it polar.
      ... in the simplest model, an electron cloud displaces into the E-field to cancel that external field, for (say) 4/5 of the atom size.
      ... that way, the external field is only NOT cancelled for 1/5 of the atom length, so the average E-field is only 1/5 of the expected intensity.

Second moment (quadrupole) : If you take two (identical) dipoles , flip one of them , and displace them from one another
. . . like this ( − + ) ( + − ) , their dipole moment will add to zero ... or like this ( + − ) ( − + ) ,
      we can describe their charge distribution by analogy with rotational "moment of inertia" as
=> Σ ( Q_i x_i² ) ... left one is negative , right one is positive ... which charge is farther from the center?
      the two dipoles' Electric field contributions almost cancel ... called an electric "quadrupole"
    => at large distances, any quadrupole's Electric field weakens as 1/r⁴ ... not   1/r³ ... see the pattern?
. . . those two opposing dipoles repel each other (use Coulomb!), and are also unstable in their orientation (angle) ...
      ( − + )       the top dipole (on this line) attracts the lower line's dipole       ( + − )
      ( + − )       and their orientation finds a stable equilibrum angle       (− + )
            . . . or are they ±y-dipoles that are displaced in the x-direction?
      ... but Σ ( Q_i x_i y_i ) = 0 , for both ... need matrix math to describe these !

#1.f: Charge Density - - - - - - - - - - - - charge spread over a region of space
      pretending that charge does not come in tiny lumps on the atomic scale, treat as a "continuous fluid" (a gas)

Volume charge density = Q/Volume = ρ_Q ... lowercase greek rho (not latin "pee") same as mass density
      => Q_total = ρ_average V_total . . . just like mass = "mass density" · Volume

Surface charge density = Q/Area = σ ... lowercase greek sigma (their "s" , abbreviation for "surface")
      => Q_total = σ_average A_total . . . just like population = "population density" · Area

Linear charge density = Q/Length = λ ... lowercase greek lambda ("l" for "length")
      => Q_total = λ_average L_total . . . (like # cars = traffic density · road length)

#2.b: Gauss' Law - - - - - - - - - - - - - -
When charge is spread out , or at least the E-field is spread out, it may be calculated (or estimated) using Gauss :
. . . Gauss re-words Coulomb: E r² = k_C Q ... but the Area of the shell that E pierces is   4π r²
=> ΣE_piercing·A_outward = 4πk_C ΣQ_inside .

imagine a box totally enclosing some charge inside its "Gaussian surfaces" ; some of its surface Area will be pierced by E
. . . (E·A is called "Electric Field flux" (analogous to the fluid flux =ρv·A) even tho the E-field does not flow)
      choose your box shape so that E pierces thru one surface Area Normally (i.e,  |  to its Length and to its Width)
      . . . choose a shape so that E scrapes along other surfaces (parallel to their L or W, so none pierces => E_piercing·A_outward = 0 ) ;
      (the box is usually shaped like the conducting surface that the charges are spread-out on)
      E-field piercing an Area inward is counted as negative E·A outward .
. . . 4πk_C "weights" the enclosed charge ; write Gauss'Law with each Electric field labeled by the surface it pierces
. . . write each Area as the appropriate function of geometry (rectangle L×W , disk πR×r , cylinder 2πR×L , shell 4πR² )
. . . then solve for your unknown E , or your unknown Q_enclosed , or your unknown radius or length.

For a hypothetical "point charge", Gauss's Area is a spherical shell outside that charge
. . . The E-field pierces the Gaussian shell along the Normal everywhere on the shell
      it looks identical to what a charged ball's E-field would look like
. . . write Gauss as E 4 π r ² = 4 π k_C Q_inside ... for either case.

Two other special-case results:
long straight wire , length L covered uniformly with charge Q: E inside the wire is zero
. . . outside the wire (and outside the charge Q) , at radius r from wire center, on a cylinder shape , E satisfies
      E 2πr L = 4πk Q . . . =>   E = 2 k Q/Lr (outward) ... notice: 1/r ! (not squared)

large thin flat plate , Area A , covered uniformly with charge Q: E inside the plate is zero
. . . "close to" the plate, above and below it, E satisfies
      E (A_top + A_bottom) = 4 πk Q . . . =>   E = 2πk Q/A(away) ... notice: almost uniform !

We can make the E-field more uniform (and twice as strong) by having a positive plate surface "close to" the negative plate
      "close to" means the plates are parallel, uniform distance d apart (along their shared normals)
. . . then, all the E-field from the positive charges goes thru the gaussian Area between the plates
=> E_{dipole plates} = 4 π k Q_plate / A_plate . . . if the gap between the plates is air ≈ vacuum
. . . if the plates are held apart by a polarizable dielectric (say, plastic) the E-field in the gap is weaker by factor   κ
      => κ E_{dipole plates} = 4 π k_C Q / A_{each plate} . . . in topic 2 we will replace   4 π k_C   with   ¹/ε₀
=> κ ε₀ EċA_{Cap_plate} = Q_plate ... this is the standard Capacitor setup.

Putting example numbers to the conductor E=0 paragraph:
. . . let's put a 2" copper cube in a fairly stiff −16,000 N/C (^x) E-field
. (a) how much charge is on each face?
      The left-pointing E-field ends on the right-pointing 0.05 [m] × 0.05 [m] = 0.0025 [m²] Area => E·A = −40 [Nm²/C] as the E-field "flux".
      4 π k_C Q = −40 [Nm²/C] => implies that the flux source   k_C Q = −10/π [Nm²/C] = −3.18 [Nm²/C] .
    => Q = −3.18/9E9 [1/(1/C)] = −3.54E-10 [C]   = −35 [nC] . . . an equal amount of + charge is on the left face.
. (b) How many electrons moved to the right face?
      Q = N e => N = −35E-9 [C]/1.6E-19[C] = −2.21E+9 ... "lacking protons = excess electrons"
. (c) How many electrons per copper atom on that right face?
      atoms are roughly 0.3 [nm] apart, so occupy 0.09E-18 [m²] Area each => N_Cu = 0.0025 [m²] / 0.09E-18 [m²] = 2.8E16 atoms on the face.
    => only 2.21E+9 (=2.2 billion) out of 28 million billion even have an extra electron ... 2 out of 25 million ... 0.08 ppm
      the extra electrons are √(25E6) = 5000 atoms apart.
. (d) well, how long would it take one of these rare electrons to "shift" one atom diameter toward the surface (0.3 nm) ?
      in a −16,000 [N/C] E-field, F = q E = (−1.6E-19 [C])(−16,000 [N/C])(^x) = 2.56E-15 [N] (^x) .
      F = m a => a = F/m = 2.56E-15 [N] / 0.911E-30 [kg] = 2.81E+15 [m/s²] . . .
      Δx = ½ a t² => t = √(2 Δx/a) = √(0.6E-9[m]/2.81E+15 [m/s²]) = √(2.14E-26) = 1.46E-13 [s]   = 146 [fs]
. (e) What Force is applied to the left face? The right face? Net Force? Do these Forces effect the copper cube?
      right face's −3.54E-10 [C] is in a −16,000 [N/C] E-field, so F = q E = (−3.54E-10 [C])(−16,000 [N/C])(^x) = 5.52E-6 [N] (^x) , pulled rightward.
      left face is pulled leftward just as hard ... cube feels 0 total electric Force, but does experience minor stretch due to 5 micro-Newtons Tension.

(note: that example did not use vector trig.)
.