Work done in a force field

1
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

2
example movement in a line-charge field

• path constant r, around an arc of a circle
• obviously zero!
• path constant ?, from r b to r a

• check the sign!
• we went FROM b TO a!

3
Conservative fields

• if the work done is independent of the path taken
  the force field is conservative
• in such a case, this suggests that the work
  integral might be a useful characteristic of the
  field
• define the potential difference V as the work
  done in moving a unit positive charge from point
  B to point A in the field E the order of
  integration will take care of the signs!
• this is a SCALAR quantity!
• example for the line charge we did on the last
  page we would have

• check the sign!
• for rA gt rB
• for rA lt rB

4
Example line charge again

• lets try a complicated path from the point B at
  (xB, yB) to the point A at (xA, yB)
• first travel from (xB, yB) to (xB, 0), the
  direction is the ay direction, along a path with
  x constant,
• dl dy(ay)
• now travel from (xB, 0) to (xA, 0), the direction
  is the ax direction, along a path with y
  constant,
• dl dx(ax)
• finally travel from (xA, 0) to (xA, yA), so the
  direction is the ay direction, along a path with
  x constant,
• dl dy(ay)
• the order of integration takes care of all the
  signs!

5
Example line charge again

• first part of complicated path from the point B
  at (xB, yB) to the point A at (xA, yB)
• straight from (xB, yB) to (xB, 0)

6
Example line charge again

• second part of complicated path from the point B
  at (xB, yB) to the point A at (xA, yB)
• straight from (xB, 0) to (xA, 0)

7
Example line charge again

• third part of complicated path from the point B
  at (xB, yB) to the point A at (xA, yB)
• straight from (xA, 0) to (xA, yA)

8
Example line charge again

• complicated path from the point B at (xB, yB) to
  the point A at (xA, yB)
• the whole thing!

• this is exactly what I got before by just
  integrating directly along r
• MUST be true because the electrostatic field is
  conservative

9
Potential references

• potential difference is clearly specific to the
  beginning and ending points
• for a single point charge at the origin consider
  a path along a radial direction FROM B TO A
• potential uses a reference point for
  calculation
• the location of the reference is somewhat
  arbitrary
• one logical reference location is infinity,
  especially for something like a point charge

• check the sign!

10
Potential for a charge distribution

• consider a point charge located at r1 (not at the
  origin)
• the potential must still only depend on the
  distance between the observation point and the
  location of the point charge
• note this assumed the reference was zero
  potential at infinity
• since Coulombs law is linear the potential of
  many charges should add
• and for a charge distribution

this is somewhat easier to evaluate than the
vector sum/integral needed to calculate the field
11
Potential references

• potential uses a reference point for
  calculation
• for charge distributions that occupy a finite
  region of space using zero at infinity usually
  works well
• another possible choice other than infinity
• lets just call the reference value Vb at point
  rB

12
Example line charge

• for a uniform line charge density located on the
  z-axis

13
Example line charge

• for a uniform line charge located on the
  z-axis
• now we need VB
• clearly its the same integral, just change to rB

14
Example line charge

• for a uniform line charge located on the z-axis

15
Equipotential surfaces

• an equipotential surface (or more simply, an
  equipotential) is such that if the beginning and
  end points fall on that surface, the potential
  difference is zero
• if E?dl is zero everywhere on paths on the
  surface this would work
• E perpendicular everywhere to the surface seems
  like it would work!
• single point charge spheres centered at the
  charge
• uniform spherical charge distributions spheres
  centered on the distribution
• infinitely long line charge cylinders centered
  on the line charge

16
Gradients

• consider the potential difference between two
  points that are very close together
• lets try this in rectangular coordinates
• but from calculus the definition of a total
  differential for any function is

17
Gradients continued

• so comparing
• hence we have
• we define the gradient as this set of
  differential operations on a scalar function
• the interpretation is
• magnitude is the maximum rate of change of the
  scalar function at the point of observation
• the direction is the direction you must travel to
  see this maximum rate of change

18
The gradient operator

• for convenience we can write
• or, for convenience, we have the vector operator
  grad
• and finally
• electric field E is equal to the gradient of the
  scalar potential
• wed read this this equation as
• E equals minus grad V
• or E equals minus del V

19
Gradient in various coordinate systems

• rectangular
• spherical
• cylindrical

20
Example line charge (again!)

• let our zero potential reference occur at the
  point (xb, yb)
• given the cylindrical symmetry of this problem,
  the circle of radius rb (xb2 yb2)1/2 would
  clearly be an equipotential, so all points on
  that circle would also be at V 0
• then the potential at a point A _at_ (x, y) would be
  given by
• we already did this by integrating over the
  charge distribution
• or in rectangular coordinates

21
Now calculate E from V for line charge

• in rectangular coordinates

22
Now calculate E for line charge from V

• in cylindrical coordinates

23
Potential produced by the electric dipole

• lets consider two point charges, of equal but
  opposite charge
• we can get the distance using the law of cosines

r
q
d
r-
d
24
The electric dipole potential

• lets consider the case when the observation
  distance r gtgt d

25
The electric dipole

• observation distance r gtgt d

26
The electric dipole

• traditional to define the dipole moment
• then recalling that cos q z?r

27
The electric dipole using the gradient to find
electric field from potential

• observation distance r gtgt d

28
The electric dipole potential and electric field

• observation distance r gtgt d

z
x

• observation on the z-axis q 0

• observation on the x-axis q p/2

• down and perpendicular to the x-axis

• up and along the z-axis

29
electric dipole equipotentials

• observation distance r gtgt d

• for a specified value of potential Vo, can we
  trace out an equipotential line?

• so for a specified value of potential Vo, an
  equipotential line is one such that r varies as a
  function of q!
• in rectangular coordinates

30
electric dipole equipotentials

• for a specified value of potential Vo, we can
  trace out an equipotential line using q (0 ? p)
  as the parameter

• then start all over with a new value of potential

• pick a value of potential Vo, then pick a value
  of q, then you have x and z

• now pick a new value of q, get the new x and z,
  repeat for all values of q

• calculation

31
The electric dipole electric field stream lines

• stream lines

32
The electric dipole electric field from the
potential using the gradient

• stream lines

• calculation

33
The electric dipole how big is the field?

• recall that the field is related to the gradient
  of potential
• in the direction of maximum change in potential
• lets look at our drawing with equipotential
  lines
• drawn with equal steps in V
• closely spaced potential lines indicate larger
  fields!

V3
V2
drC
drA
EA
EC
drB
V1
EB
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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.

35
What have we so far?

• electric field from charge distributions
• Coulombs law
• relation between flux and charge
• Guasss law
• work and potential from charge distributions
• relation between potential and field

36
Whats next?

• Id still like to understand how signals move
  (propagate) through a long telegraph cable
• to do that we need a better understanding of
  materials
• conductors
• dielectrics