unit 9: physical examples of fields. potential energy function. unit...
TRANSCRIPT
Unit 9: Physical examples of fields.Potential energy function.
Unit 10: Gradient of potential.Gradient as directional derivative.
Units 9 & 10 are on overlapping topics. We shall develop them together.
PCD-08
Equipotentials and Field Lines of an electric dipolePCD-08
Primary, elementary entities in Physics:
Physical Universe is made up of Particles and Fieldsand their mutual interactions
Particles: Material world
Fields: ‘Action at a distance’
Fields PotentialsIntegration / Constant of IntegrationBoundary Value problem
DifferentiationPCD-08
Scalar/ Vector Fields: ‘Point’ function
( ) ( , , ) ( , , ) ( , , )
( ) ( , , ) ( , , ) ( , , )
r x y z r z
A A r A x y z A r A z
ψ ψ ψ ψ θ ϕ ψ ρ ϕ
θ ϕ ρ ϕ
= = = =
= = = =
r
ur ur ur ur urr
Examples of Scalar Point Functions
• temperature• gravitational/electrostatic
potentials• pressure in a liquid column
Examples of Vector Point Functions
• velocity field• electric field• magnetic field
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Scalar/ Vector Fields: ‘Point’ function
Functions of ‘space’ and ‘time’
( , ) ( , , , ) ( , , , ) ( , , , )
( , ) ( , , , ) ( , , , ) ( , , , )
r t x y z t r t z t
A A r t A x y z t A r t A z t
ψ ψ ψ ψ θ ϕ ψ ρ ϕ
θ ϕ ρ ϕ
= = = =
= = = =
r
ur ur ur ur urr
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0 0
The rate of change of with distance is a scalar given by
( ) ( ) ( )lim lims s
s
d r r r rds s sδ δ
ψ
ψ δψ ψ δ ψδ δ→ →
+ −= =
r r r r
DIRECTIONAL DERIVATIVE is a SCALAR QUANTITY
which has a DIRECTIONAL ATTRIBUTE.
This ‘rate’ (‘slope’) depends on the direction in which the displacement is considered.
r
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δ r
Ratio of two scalar quantities.
It has a ‘directional attribute’
0 0
=
( ) (
) lim lim
=
s s
x y zx y z
z
d r
z
r
d s
r
r rs sδ δ
ψ ψ ψδ δ δ
ψ ψ ψδρ δϕ δρ ϕ
ψ ψ ψδ δ
ψ δψ ψ δ ψδ δ
ϕ
δψ
θ δϕθ
→ →
∂ ∂ ∂+ +
∂ ∂ ∂
∂ ∂ ∂+ +
∂ ∂ ∂
∂ ∂ ∂= + +∂ ∂
+ −= =
∂
r r r
Cartesian Coordinate System
Cylindrical Polar Coordinate System
Spherical PolarCoordinate System
(r)= ( , , )
(r)= ( , , )
(r)= ( , , )
x y z
z
r
ψ ψ
ψ ψ ρ ϕ
ψ ψ θ ϕ
r
r
r
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0
=
The directional derivative( ) ( ) lim
=
s
dx dy dzx ds y ds z ds
d d dzds ds z ds
dr d dr ds
d
d
r r rds
s ds
sδ
ψ ψ ψ
ψ ρ ψ ϕ ψρ ϕ
ψ ψ δ ψ
ψ ψ θ ψϕ
δ
ϕθ
→
∂ ∂ ∂+ +
∂ ∂ ∂
∂ ∂ ∂
+ −=
+ +∂ ∂ ∂
∂ ∂ ∂= + +∂ ∂ ∂
r r r
PCD-08
Cartesian Coordinate System
Cylindrical Polar Coordinate System
Spherical PolarCoordinate System
These expressions can be written very nicely by using the expressions for the displacement in various coordinate systems.dr
uur
(r)= ( , , )
(r)= ( , , )
(r)= ( , , )
x y z
z
r
ψ ψ
ψ ψ ρ ϕ
ψ ψ θ ϕ
r
r
r
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ˆ ˆ ˆr =e e e zd d d dzρ ϕρ ρ ϕ+ +r
ˆ ˆ ˆr =e e ex y zd dx dy dz+ +r
ˆ ˆ ˆr = e e ex y zx y z+ +r
ˆ ˆr = e e zzρρ +r
ˆr =re rr ˆ ˆ( ) ( )r rdr dr e r de= +
r
ˆ ˆˆ( ) r rr
e edr dr e r d dθ ϕθ ϕ
⎡ ⎤∂ ∂= + +⎢ ⎥∂ ∂⎣ ⎦
r
ˆ ˆ ˆsinrdr dre rd e r d eθ ϕθ θ ϕ= + +r
ˆ ˆ ˆr =( )e ( e ) ( )e zd d d dzρ ρρ ρ+ +r
ˆ ˆ ˆr =e e ex y zd dx dy dz+ +r
ˆ ˆ ˆ sinrdr e dr e rd e r dθ ϕθ θ ϕ= + +r
Position and Displacement vectors in various coordinate systems
In order to avoid making careless mistakes, always try to write unit vectors first, differential elements last!
Example:
0
( ) ( )lim
dx dy dz = + +
x ds ds ds
s
d r r rds s
dds y z
δ
ψ ψ δ ψδ
ψ ψ ψ ψ
→
+ −=
∂ ∂ ∂∂ ∂ ∂
r r r
ˆ ˆ ˆ=e e e
ˆ ˆ ˆ= e e e
x y z
x y z
x y z
x y zψ ψ
∂ ∂ ∂∇ + +
∂ ∂ ∂
⎡ ⎤∂ ∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
ur
s 0
ˆ
ˆ lim
, tiny
increament
, differential
increament
d uds
r drus ds
s r
ds dr
δ
ψ ψ
δδ
δ δ
→
= •∇
= =
=
=
r
uur uur
uur
uur
ˆ ˆ ˆr =e e ex y zd dx dy dz+ +r
Gradient in the Cartesian Coordinate System
PCD-08
( ) 1ˆ ˆ ˆ ˆ ˆ ˆe e e e e ez zzzρ ϕ ρ ϕ
δψ
δρ ρδϕ δ ψρ ρ ϕ
=
⎛ ⎞∂ ∂ ∂+ + • + +⎜ ⎟∂ ∂ ∂⎝ ⎠
:
ˆ ˆ ˆr =e e e
:
z
How should we definesuch that
r
zwhere
ρ ϕ
ψ
δψ δ
ρ ϕ
ψ
δ δ ρδ δ+
∇
= • ∇
+
ur
uur ur
r
1ˆ ˆ ˆ=e e
e
!
z
Following form of thegradient operator will work
zρ ϕρ ρ ϕ∂ ∂ ∂+ +
∂ ∂∇
∂
ur
( )
ˆ ˆ ˆe e e z
z
r
z
z
ρ ϕ
δ
δψψ ψ ψ
ψ
δρ ρδϕ δ
δρ δϕ δρ ϕ
ψ
= •
=∂ ∂ ∂
= + +∂ ∂
∇
+ + • ∇
∂uur ur
ur
1Note how the
cancelsρρ
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Cylindrical Polar Coordinate System
( ) ( ) ( ) 1 1ˆ ˆ ˆ ˆ ˆ ˆe e e sin e e esinr rr r r
r r rθ ϕ θ ϕ
δψ
δ δθ θ δϕ ψθ θ ϕ
=
⎛ ⎞∂ ∂ ∂⎡ ⎤+ + • + +⎜ ⎟⎣ ⎦ ∂ ∂ ∂⎝ ⎠
( ) ( ) ( )ˆ ˆ ˆr =e e e
s
:
:
in r
How should we definesuch that
r
rwher
re
rθ ϕδ δ δθ θ δ
δ ψ
ϕ
ψ
ψ δ
+ +
∇
= • ∇
ur
uur
r
ur
1 1ˆ ˆ ˆ=e e es
i
!
nr
Following form of thegradient operator wil
r r
l wo k
r
r
θ ϕθ θ ϕ∂ ∂ ∂+ +
∂∇
∂ ∂
ur
( ) ( ) ( )
ˆ ˆ ˆe e e sin r
r
r r
rr
rθ ϕ
δ ψ
δ δθ
δψψ ψ ψδ δθ
θ δϕ
δϕϕ
ψ
θ
= •
=∂ ∂ ∂
= + +
∇
⎡ ⎤+ + • ∇⎣
∂ ∂ ∂
⎦
uur ur
ur
PCD-08
Note the cancellation of the factors that are circled
Spherical Polar Coordinate System
0
( ) ( )lim
=
s
d r r rds s
d d d dzds ds dss zd
δ
ρ ψ
ψ ψ δ
ϕ ψ ψρ ϕ
ψδ
ψ
→
∂ ∂ ∂+ +
∂
+ −=
∂ ∂
r r r
s 0
ˆ
ˆ lim
, tiny
increament
, differential
increament
d uds
r drus ds
s r
ds dr
δ
ψ ψ
δδ
δ δ
→
= •∇
= =
=
=
r
uur uur
uur
uur
Directional Derivative in the Cylindrical Polar Coordinate System
ˆ ˆ ˆr =e e ezd d d dzρ ϕρ ρ ϕ+ +r
1ˆ ˆ ˆ=e e e
1ˆ ˆ ˆ= e e e
z
z
z
z
ρ ϕ
ρ ϕ
ρ ρ ϕ
ψ ψρ ρ ϕ
∂ ∂ ∂∇ + +
∂ ∂ ∂
⎡ ⎤∂ ∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
ur
PCD-08
0
( ) ( ) lim
=
s
dr d
d r r rds s
d dr ds dds s ds
δ
ψ ψ δ ψδ
ψ ψ ψ θ ψ ϕθ ϕ
→
+
∂ ∂ ∂= + +∂ ∂ ∂
−=
r r r
s 0
ˆ
ˆ lim
, tiny
increament
, differential
increament
d uds
r drus ds
s r
ds dr
δ
ψ ψ
δδ
δ δ
→
= •∇
= =
=
=
r
uur uur
uur
uur
ˆ ˆ ˆ sinrdr e dr e rd e r dθ ϕθ θ ϕ= + +uur
1 1ˆ ˆ ˆ=e e esin
1 1ˆ ˆ ˆ= e e esin
r
r
r r r
r r r
θ ϕ
θ ϕ
θ θ ϕ
ψ ψθ θ ϕ
∂ ∂ ∂∇ + +
∂ ∂ ∂
⎡ ⎤∂ ∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
ur
PCD-08
Directional Derivative in the Spherical Polar Coordinate System
1 1ˆ ˆ ˆ=e e esinr r r rθ ϕθ θ ϕ
∂ ∂ ∂∇ + +
∂ ∂ ∂
ur
1ˆ ˆ ˆ=e e ez zρ ϕρ ρ ϕ∂ ∂ ∂
∇ + +∂ ∂ ∂
ur
ˆ ˆ ˆ=e e ex y zx y z∂ ∂ ∂
∇ + +∂ ∂ ∂
ur
s 0
ˆ
ˆ lim
, tiny
increament
, differential
increament
d uds
r drus ds
s r
ds dr
δ
ψ ψ
δδ
δ δ
→
= •∇
= =
=
=
r
uur uur
uur
uur
PCD-08
ˆ ˆx yT e e∇ = +r
T = 20 + x + y
T = 100 + x y
This picture shows the value of scalar function T;
Brighter regions have larger values of T
ˆ ˆy xT xe ye∇ = +r
T = 100 + x y
Example:
Map of a scalar field
fields -- gradient
This picture shows the value of scalar function T;
Brighter regions have larger values of T
2 2
Equipotential surf1
c00
a eT x y= =
2 2 400T x y= =
fields -- gradient
We can visualize several equipotential surfaces for different values of Te.g. countours
fields -- gradient
Example
Φ(x,y,z) = x2 +y2+z2
level surface Φ =c ( a positive constant) is a sphere of radius √c centered at the origin.
∇ Φ = 2x ex + 2y ey+ 2 zez = 2 r, along er
gradient at a point with position vector r is perpendicular to the equipotential at r
Divide gradient a point by its magnitude to get the unit vector
The shades of grey represent the intensity of the scalar field; the blacker, the more intense.The blue vectors indicate the directions of the gradient.
Example 1 Example 2 : uniform gradient
a) T = 20 + x + y
b) T= 100+xy
c) T = x2y2
Examples(calculation of gradient and visualization of the fields)
ˆ ˆx yT e e∇ = +r
ˆ ˆy xT xe ye∇ = +r
2 2ˆ ˆ2 2x yT xy e x ye∇ = +r
fields -- gradient
Example
Φ(x,y,z) = x2 +y2+z2
level surface Φ =c ( a positive constant) is a sphere of radius √c centered at the origin.
∇ Φ = 2x ex + 2y ey+ 2 zez = 2 r, along er
T = 15 + ρ cos φ
Geostationary Orbits ⊂ Geosynchronous
Orbits Geostationary Orbit:The satellite orbits the earth at exactly the same speed as the earth turns and
at zero latitude (equator). It appears at the same zenith point from the equatorial point underneath it.
Geosynchronous Orbit:The satellite’s orbit is synchronized with the earth's rotation. The plane of the
satellite’s orbit may be tilted with respect to the equatorial plane. The satellite will then appear to move around along a latitude even if it will remain at the same longitude.
Many advantages! We need detailed maps of ‘potentials’ and ‘fields’ to launch such objects in the sky.
WMAP: Wilkinson Microwave Anisotropy ProbeOrbits around the Lagrange point L2 of the Sun-Earth system.
The Lagrange points are analogous to the points where geosynchronous orbits can be launched, but with regard to a 3-body problem.
At the Lagrange points, the resultant gravitational pull of the sun and the earth on an object, such as a satellite, will precisely equal the centripetal force that is required for that object to rotate with them. At the Lagrange points, net force ~ zero!Lagrange found five such points (‘LAGRANGE POINTS’) for the sun-earth system using the PRINCIPLE OF LEAST ACTION.
SOHO
Solar and HeliosphericObservatory
WMAP
WilkinsonMicrowaveAnisotropy Probe
The L1 point of the Earth-Sun system provides uninterrupted view of the sun.
At L2: Sun, Earth, & Moon are
always behind the instrument's field of view.
http://map.gsfc.nasa.gov/mission/index.html
http://en.wikipedia.org/wiki/Lagrangian_point
http://map.gsfc.nasa.gov/media/990533/index.html
Colonizing space?
At Lagrange points?
Wilkinson Microwave Anisotropy Probe: WMAP
The average temperature is 2.725 Kelvin (degrees above absolute zero; equivalent to -270 C or -455 F), and the colors represent the tiny temperature fluctuations, as in a weather map.
Red regions are warmer and blue regions are colder by
about 0.0002 degrees.
The cosmic microwave temperature fluctuations from the 5-year WMAP satellite data seen over the full sky.
http://map.gsfc.nasa.gov/news/index.html
MEASURING COSMIC ASYMMETRY?
http://www.aerith.net/comet/catalog/1998J1/pictures.html
Roughly eighty-five percent of the SOHO discoveries, and also this one, are fragments from a once great comet that split apart in a death plunge around the Sun, probably many centuries ago. The fragments are known as the Kreutz group and now pass within 1.5 million kilometres of the Sun's surface when they return from deep space.
The Kreutz-Group comet SOHO-1500 was spotted on June 25th 2008 in images taken by the LASCO C2 coronagraph.
http://soho.nascom.nasa.gov/hotshots/2008_06_23/1500thcomet.tif
Most successful comet catcher in history. SOHO has discovered its 1500th comet, making it more successful than all the other discoverers of comets throughout history put together.
SOHO: spacecraft that was designed as a solar physics mission. SOHO's history-making discovery was made on June 25th 2008 by US-based amateur astronomer Rob Matson. This is Rob's 76th SOHO comet find.
SOlar and Heliospheric Observatory has one big advantage over everybody else: its location.
Situated between the Sun and the Earth, it has a privileged view of a region of space that can rarely be seen from Earth.
From the surface of the planet, the space inside our orbit is largely obscured because of the daytime sky and so we only clearly see close to the Sun during an eclipse.
http://www.physics.montana.edu/faculty/cornish/lagrange.html
[1] WMAP uses the moon to get a gravity assisted boost on to the L2 point. This saves a lot of fuel that can be used later to extend WMAP's operational life. This animation shows the flight path around the Moon. Credit: NASA / WMAP Science Team http://map.gsfc.nasa.gov/media/990534/990534_320.mp4
[2] WMAP Flight to L2 WMAP orbits around the L2 Lagrange point, one million miles beyond Earth. The sun shield/solar panels always protect it from the radiation generated by the Sun. This animation Shows the flight path from Earth to L2 http://map.gsfc.nasa.gov/media/990533/990533_320.mp4[3] SOHO’s orbit around the sun and around L1 http://www.youtube.com/watch?v=cGm-nrgAIzM&feature=related
We shall now show three video clips, two from NASA, and one from youtube
Understanding gradient
Electric field and potential inside a conductor
Conductors have `free electrons’
Consider charges to be placed on a conducting solid/hollow sphere
Electrostatic field inside a conductor is zero
Any net charge resides on the surface only, not inside
A conductor is an equipotential
For any two points and inside the conductor,
( ) ( ) . 0 and hence ( ) ( )b
a
a b
U a U b E dl U a U b− = − = =∫rr
The field lines just outside the conductor are perpendicularto the surface of the conductor
Ref: Introduction to Electrodynamics, Griffiths
Understanding gradient
Field and potential due to a point dipole
2 20
3
3
2 cos=
1 sin=
1 =0sin
=
=
=
Field is the negative gradient
The elctrostatic potential due to a point dipole is cos 1( ,
of potential
, ) = where 4
Thus
U kpr rU kp
r rU
r
r
k r p kprU r kr r
E
E
E
θ
ϕ
θ
θθ
θ ϕ
θθ ϕπε
∂∂
∂∂
∂∂
−
−
−
⋅= =
r r
3 ˆ ˆ( , , ) ( 2cos sin )rkpE r e er θθ ϕ θ θ= +
r
pr
θcosp θsinp θ
z
Ref: Introduction toElectrodynamics, Griffiths3
ˆ ˆsince cos siˆ ˆ3( )
nr
r rpp e p e
e e pp
E kr
θθ θ=
⋅ −=
−rr
r
r
1 1ˆ ˆ ˆU= e e esinr U
r r rθ ϕθ θ ϕ⎡ ⎤∂ ∂ ∂
∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
Understanding gradient
Electric field and potential due to an infinitely long wire2ˆ where is the uniform charge density per unit lengthkE eρλ λρ
=r
ˆ ˆ ; comparing coefficients of and
2 ; 0
he 12 ln some conce nstant (indep. of )
. . 2 l
n
E U e e
U k
ci e U
U k
k
Uρ ϕ
λρ
λ
λρ
ρ
ρ
ϕ
ρ
= −∇
∂ ∂= − =
⎧ ⎫
∂
= +⎨ ⎬⎩ ⎭
⎧ ⎫= ⎨
⎭
∂
⎬⎩
r r
1ˆ ˆ ˆU= e e ez Uzρ ϕρ ρ ϕ
⎡ ⎤∂ ∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
Understanding gradient
Obtaining the field for a given potential
1 0 2 2
2 sinconsider an electrostatic potential tan2
aU Ua
π ρ ϕρ
−⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟= +⎨ ⎬
⎜ ⎟−⎪ ⎪⎝ ⎠⎩ ⎭
0
2 2
2
2 sin1
024
1 sin2
0
tan at 2
2 sin2 sin
2 cos
a
aa
U U a
aUU U aE
aUE
ϕρ
ρ
ϕρ
ϕ
π ρ
ϕϕρ ρ ρ
ϕρ
⎛ ⎞−⎜ ⎟⎝ ⎠
+
⎧ ⎫= + >>⎨ ⎬⎩ ⎭
∂= − = =
∂ ⎧ ⎫⎨ ⎬⎩ ⎭
−=
1ˆ ˆ ˆU= e e ez Uzρ ϕρ ρ ϕ
⎡ ⎤∂ ∂ ∂∇ + +⎢ ⎥∂ ∂ ∂⎣ ⎦
ur
2 4( ) 2 3U x Ax Bx= −
Plot for A=B=1
Understanding gradient: Shapes of fields and potentials, equilibria
2
2
1ˆ ˆ ˆ
. . cos 2 U= cos 2 + ( , ) 2
This implies sin 2 .
Comparing now the corresponding terms of ,
: 0 ( )
Now ;
zU U UF e e e
zUi e f z
U f
Ufwe get f f z c
U f Uand zz z z
ρ ϕρ ρ ϕρρ ϕ ϕ ϕ
ρ
ρ ϕϕ ϕ
ϕ
∂ ∂ ∂= − − −
∂ ∂ ∂
∂= ⇒
∂∂ ∂
= − +∂ ∂
− ∇∂
= ⇒ = +∂
∂ ∂ ∂= = − ⇒
∂ ∂ ∂
ur
ur
2
2 2
2
( , , ) cos 22 2
zf
zU z kρρ ϕ ϕ
−=
∴ = − +
Given: Force experienced by a particle is
ˆ ˆ ˆ( , , ) cos 2 + sin 2 zzF z e e eρ ϕρ ϕ ρ ϕ ρ ϕ= − +ur
Obtaining the potential for a given field