Flux

1
Flux

• imagine a fluid flowing through a rectangular
  pipe, with velocity constant across the cross
  section of the pipe
• the flux density of fluid flowing through the
  pipe is the amount per unit area
• should be something like
• velocity/cross sectional area
• what if you choose a surface not perpendicular to
  the flow?
• shouldnt really change things
• we need to find the projection of the surface
  onto the direction of flow

2
directed surface elements

• we need to associate a direction with a surface
  (area) element when we find flux due to a vector
  field F
• the direction is outward normal to the surface
• a flux is the projection between the surface and
  the vector field A
• use a dot product!

q
adapted from http//hyperphysics.phy-astr.gsu.edu/
hbase/electric/gaulaw.htmlc3
3
Flux from a point charge

• recall that for a single point charge Q fixed at
  the origin
• what is the flux due to this charge crossing
  the surface of a sphere centered on the charge?

q
R sinq d?
R
R sinq
d?
the charge enclosed / eo
4
Flux from a point charge

• what about an arbitrarily shaped closed surface?
• area of sphere subtended by a solid angle d? is
  r2d?
• the amount of flux would be
• but the perpendicular area of a surface that
  subtends the solid angle d? is always r2d? !
• so it still works
• note this works because Coulombs law is 1/r2 !!

adapted from Purcell
5
Net flux through an empty region

• we found the flux through an arbitrary surface
  enclosing a charge Q is always Q/eo

• consider what happens as the tube connecting the
  two regions shrinks
• first sphere gives Q/eo,
• as the size of the tube gets very small, it
  contributes nothing

• the flux produced by the external charge
  passing through the second empty sphere from one
  side to the other must contribute nothing !

6
Multiple point charges Gausss Law

• since Coulombs law is linear, I suspect
  everything should just add
• each sphere gives Q/eo,
• the flux passing through one sphere produced by
  the external charge is zero
• as the size of the tube gets very small, it
  contributes nothing
• clearly this will generalize to as many charges
  as you want

7
Electric flux density D

• to get rid of the e we use a new vector quantity
  D
• aka the displacement density, displacement
  flux densityunits E is Volt/meter, e is
  Farad/meter, so D is VF/m2 Coul/m2
• using Gausss Law
• trick is to look for symmetries and pick the
  right surface
• you want to pick a surface such that D?dS (or
  equivalently, E?dS) is either
• constant
• or zero (either D itself is zero, or D is
  perpendicular to dS)

8
Field due to an infinitely long line charge

• calculating fields using a gaussian surface
• by symmetry, the field points radially outward
  from the line
• for line charge on the z-axis, field is
  cylindrically symmetric
• lets try a cylinder of radius r, height h, with
  the line charge running down its center
• sides
• bottom / top

9
Infinite charged sheet

• again we have a lot of symmetry only field
  component must be perpendicular to sheet
• gaussian surface any shape with top and bottom
  parallel to plane, sides perpendicular to plane
• lets just use a simple box
• top area element dxdy field is perpendicular
  to surface, parallel to surface normal ?
  D?dS Ddxdy ? ?D?dS Darea
• bottom area element dxdy field is
  perpendicular to surface, parallel to surface
  normal ? D?dS Ddxdy ? ?D?dS Darea
• sides field is parallel to surface,
  perpendicular to surface normal ? D?dS 0 !
• total flux 2Darea
• charge enclosed rSarea
• 2Darea rSarea ? D rS/2 ? E
  rS/2eo

10
Spherical volume charge distribution rv

• assume there is a uniform sphere of charge
  centered at the origin, radius R
• by symmetry the field must be spherically
  symmetric, with only a radially directed
  component, function of r only
• lets pick our gaussian surface to be a sphere
  centered at the origin, radius r

R
r
11
Spherical volume charge distribution rv

• for r gt R
• charge enclosed
• integral over flux density
• Gausss law

R
r
12
Spherical volume charge distribution rv

• for r lt R
• charge enclosed
• integral over flux density
• Gausss law

R
r
13
Spherical volume charge distribution rv

• so for all r

R
r
14
Notes on flux

• so to summarize, the flux over a closed surface
  indicates whether there are sources present
  inside
• lets consider what happens as the volume of the
  test region shrinks to zero

15
What about the flux emerging from a tiny volume?

• consider a box, centered at the point (x, y, z),
  in a vector field F
• magnitude of Fx on the front and back sides of
  the box
• use a Taylor series to get the approximate value
  of the function on the sides of the box using the
  value at the center
• front side of box
• back side of box

16
Flux through the surfaces of the box

• magnitude of Fx on the front and back sides of
  the box
• front side of box
• back side of box
• flux through front d? (Fx(front))(area)
• plus since Fx is same direction as outward
  surface normal on front
• flux through back d? (-Fx(back))(area)
• minus since Fx is opposite direction as outward
  surface normal on back

17
Flux through the surfaces of the box

• so total flux through the front and back sides of
  the box
• adding up all the contributions from all the
  faces gives the total flux out of the volume
  dxdydz

18
Flux through the surface of an infinitesimal
volume element

• as the volume shrinks to zero the approximation
  becomes exact
• the divergence of a vector field is the outgoing
  flux from an infinitesimal volume surrounding the
  point of evaluation
• it is a scalar operation on a vector field
• in rectangular coordinates
• divergence of F or div F or del dot F
• note divergence has units!
• inverse of coordinate you take the derivative
  wrt in our flux example its 1/distance!

19
Combining divergence with Gausss law

• lets consider the case when the vector field is
  D, the flux density
• now what if the volume dv contains a volume
  charge density rv?
• Guasss law told us that flux equals charge
  enclosed
• here the charge enclosed is dQ rvdv

• or written more compactly we have the
  differential form of Gausss law

check the units!
20
Differential form of Gausss law

• this is also called a constitutive relationship
• it relates a field property (??D) to a material
  property (rv)
• any guesses about Gausss law and time dependent
  situations?
• what if the charge enclosed inside a surface
  changed in time?
• we know electric fields are light, traveling at
  finite speed, so the flux through a surface a
  long way away couldnt change instantaneously
• what about the differential form?
• at least now the enclosure is infinitely close
  to the charge, so propagation time shouldnt be
  an issue

21
the Divergence Theorem

• Gausss law surface integral of D is charge
  enclosed
• but Q is just a volume integral of the charge
  density
• and we have
• combining gives the divergence theorem (true in
  general)

22
Example field due to an infinitely long line
charge

• right now we only have the divergence in
  rectangular coordinates
• here Dz 0, and D is radial (x-y plane)

23
Example divergence of the field due to an
infinitely long line charge

• here Dz 0, and D is radial (x-y plane)

24
Handy tips

• rho vector (cylindrical coords) in x-y
  coordinates
• radial vector (spherical coords) in x-y-z
  coordinates
• divergence in other coordinates
• cylindrical
• spherical

25
Divergence for a spherical charge distribution

• recall the field of a spherical, uniform charge
  is just

26
General web resources

• div curl demo http//www.sunsite.ubc.ca/LivingMat
  hematics/V001N01/UBCExamples/Flow/flow.html
• explanation of div and curl, with applets
  http//www.math.gatech.edu/7Ecarlen/2507/notes/ve
  ctorCalc/dcvisualize.html
• relative velocity applet http//www.math.gatech.ed
  u/7Ecarlen/2507/notes/classFiles/partOne/RelVel.h
  tml
• vector field applet http//math.la.asu.edu/kawski
  /vfa2/index.html

27
Applets showing some vector fields

• 2-d view http//www.physics.orst.edu/tevian/micr
  oscope/
• 3-d view http//www.falstad.com/vector/
• fields available http//www.falstad.com/vector/fu
  nctions.html
• 1/r single line electric field around an
  infinitely long line of charge. It is inversely
  proportional to the distance from the line.
• 1/r double lines field around two infinitely
  long conductors. The distance between them is
  adjustable.
• 1/r2 single field associated with gravity and
  electrostatic attraction gravitational field
  around a planet and the electric field around a
  single point charge.
• This is a two-dimensional cross section of a
  three-dimensional field.
• In three dimensions, the divergence of this field
  is zero except at the origin in this cross
  section, the divergence is positive everywhere
  (except at the origin, where it is negative).
• 1/r2 double field associated with gravity and
  electrostatic attraction. gravitational field
  around two planets and the electric field around
  two negative point charges are similar to this
  field.

28
Work done in a force field

• work force distance
• but when the magnitude and direction of the force
  varies with position (i.e., the force is a vector
  field) this requires some clarification
• the differential work done depends only on the
  component of force in the same direction as
  differential distance traveled
• the total work is a line integral along the path
• example work required to move a charge Q in an
  electric filed E
• why the minus sign?
• imagine we are moving a positive charge in the
  radial direction away from another positive point
  charge at the origin
• the force is outward, in the direction of our
  motion so E?dl is positive
• BUT we dont have to do the work, we actually
  gain energy from the field ? negative work