Transcript Physics 121 Fall 2002

Electricity and Magnetism
Introduction to Physics 121
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Syllabus, rules, assignments, exams, etc.
iClickers
Quest
Course content overview
• Review of vector operations
• Dot product, cross product
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Scalar and vector fields in math and physics
Gravitation as an example of a vector field
Gravitational flux, shell theorems, flow fields
Methods for calculating fields
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Copyright R. Janow - Fall 2011
Course Content
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1 Week: Review of Vectors, Some key field concepts
– Prepare for electrostatic and magnetic fields, flux...
5 Weeks: Stationary charges –
– Forces, fields, electric flux, Gauss’ Law, potential, potential energy, capacitance
2 Weeks: Moving charges –
– Currents, resistance, circuits containing resistance and capacitance, Kirchoff’s
Laws, multi-loop circuits
2 Weeks: Magnetic fields (static fields due to moving charges) –
– Magnetic force on moving charges,
– Magnetic fields caused by currents (Biot-Savart’s and Ampere’e Laws)
2 Weeks: Induction –
– Changing magnetic flux (field) produces currents (Faraday’s Law)
Thanksgiving in here somewhere
2 Weeks: AC (LCR) circuits, electromagnetic oscillations, resonance
Not covered:
– Maxwell’s Equations - unity of electromagnetism
– Electromagnetic Waves – light, radio, gamma rays,etc
– Optics
For us, Work units begin with a weekly lecture (on Friday) and ends
About a week later when homework is covered in recitation class. There are exceptions due to scheduling. Check Page 3 of
the course outline for details.
Copyright R. Janow - Fall 2011
Phys 121 – The Big Picture
Why do you need Phys 121 – Electromagnetics?
→ It is fundamental to many areas of Science and
Engineering
• Electronic circuits (including computers)
• Sensors
• Biological function
• Wireless (and wired) communications
Mechanics
Phys 111
Electromagnetics
Phys 121
“Modern” Physics
Phys 234
Typically ECE and
Physics majors
Copyright R. Janow - Fall 2011
Phys 121 – The Big Picture
Capacitors, Resistors, inductors, and Kirchkoff’s loop
laws for circuits from Phys 121 are the basics for
• Computers
Copyright R. Janow - Fall 2011
Phys 121 – The Big Picture
Electric fields, voltages, charges from Phys 121 are the
basics for medicine and Biology
• Electro-cardiography
• biological function of Cells
Copyright R. Janow - Fall 2011
Phys 121 – The Big Picture
Electric fields, voltages, charges from Phys 121 are the
basics for Civil Engineering infrastructure
• Power Grid
• Sensors
Copyright R. Janow - Fall 2011
Phys 121 – The Big Picture
Electric fields, voltages, charges, circuits from Phys 121
are the basics for Electrical Engineering
WiFi
3G, 4G, 5G…….
lasers
Copyright R. Janow - Fall 2011
Electricity and Magnetism
Lecture 01 - Vectors and Fields
Physics 121
Review of Vectors :
• Components in 2D & 3D. Addition & subtraction
• Scalar multiplication, Dot product, vector product
Field concepts:
• Scalar and vector fields
• How to visualize fields: contours & field lines
• “Action at a distance” fields – gravitation and electro-magnetics.
• Force, acceleration fields, potential energy, gravitational potential
• Flux and Gauss’s Law for gravitational field: a surface integral of
gravitational field
More math:
• Calculating fields using superposition and simple integrals
• Path/line integral
• Spherical coordinates – definition
• Example: Finding the Surface Area of a Sphere
• Example: field due to an infinite sheet of mass
8
Copyright R. Janow - Fall 2011
Vector Definitions
- Experiments tell us which physical quantities are scalars and vectors
- E&M uses vectors for fields, vector products for magnetic field and force
Representations in 2 Dimensions:
y

A
• Cartesian (x,y) coordinates

A  A x î  A y ĵ
A
A 
A 2x  A 2y
-1 A y 
  tan 

 Ax 
• Addition and subtraction
of vectors:
• Notation for vectors:
Copyright R. Janow - Fall 2011

ĵ
• magnitude & direction
k̂
Ay = A sin()
î
Ax = A cos()
x
z



C  A  B means Cx  A x  Bx and Cy  A y  By


C  - A means Cx   A x and C y   A y
F  ma


F  ma
F  ma
Vectors in 3 dimensions
•
Unit vector (Cartesian) notation:
•
Spherical polar coordinate representation:
1 magnitude and 2 directions
Rene Descartes
1596 - 1650

a  a x î  a y ĵ  a zk̂

a  (a, , )
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z
Conversion into x, y, z components
a x  a sin  cos 
a y  a sin  sin 
a z  a cos 
•

a
az
Conversion from x, y, z components

a  a 2x  a 2y  a 2z
1
  cos a z / a
  tan  1 a y / a x
ax
y

x
Copyright R. Janow - Fall 2011
ay
Definition: Right-Handed Coordinate Systems
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We always use right-handed
coordinate systems.
In three-dimensions the righthand rule determines which
way the positive axes point.
Curl the fingers of your
RIGHT HAND so they go from
x to y. Your thumb will point
in the positive z direction.
z
y
x
This course will use many right hand rules related to this one!
Copyright R. Janow - Fall 2011
Right Handed Coordinate Systems
1-1: Which of these coordinate systems are right-handed?
A.
B.
C.
D.
E.
I and II.
II and III.
I, II, and III.
I and IV.
IV only.
y
y
I.
II.
x
z
z
x
x
x
III.
IV.
y
z
Copyright R. Janow - Fall 2011
z
y
Vector Multiplication
Multiplication by a scalar:

sA  sA x î  sA y ĵ

A

sA
vector times scalar  vector whose length is multiplied by the scalar
Dot product (or Scalar product or Inner product):

B

- vector times vector  scalar
- projection of A on B or B on A

A
- commutative
A B
 ABcos( )  B  A  Ax Bx  A y By  A z Bz
Copyright R. Janow - Fall 2011
iˆ ˆj  0, ˆj kˆ  0, ˆi kˆ  0
ˆi ˆi  1, ˆj ˆj  1, kˆ kˆ  1
Vector multiplication, continued
Cross product (or Vector product or Outer product):
-vector times vector  another vector
perpendicular to the plane of A and B
- draw A & B tail to tail, right hand rule shows direction of C
  
 
C  A  B  - B  A (not commutative)


magnitude : C  ABsin( )
B
A



where

is
the
smaller
angle
from
A
to
B

C
- If A and B are parallel or the same, A x B = 0
- If A and B are perpendicular, A x B = AB (max)
  
 
 
distributiv e rule : A (B
Algebra:
  C)   A  B  A  C
associative rules : sA
  (sB)
 B
  (sA)  B  A
( A  B)  C  A  (B  C)
î  ĵ  k̂, ĵ  k̂  î , î  k̂  - ĵ
Unit vector
representation: î  î  0, ĵ  ĵ  0, k̂  k̂  0
 
A  B  (Ax î  Ay ĵ  A zk̂)  (B x î  By ĵ  Bzk̂)
 (A yBz - AzBy ) î  (A zBx - AxBz )ĵ  (A xB y - AyBx )k̂


  
 

Applications:   r  F
L  r p
F  qE 
Copyright R. Janow - Fall 2011
i
j
k
 
qv  B
Example:
A force F = -8i + 6j Newtons acts on a particle with position vector r = 3i + 4j meters relative
to the coordinate origin. What are a) the torque on the particle about the origin and b) the
angle between the directions of r and F.
  
a) Use:  r  F

  r  F  ( 3 î  4 ĵ)  ( 8 î  6 ĵ)
  ( 3x 8) î  î  ( 3x 6) î  ĵ  (4 x 8) ĵ  î  (4x 6) ĵ  ĵ
 18 k̂  32 k̂
b)
 ˆ  50 k̂ N.m

Use: |  |  r F sin( )
r  [ 32  42 ]1 / 2  5
ˆ  50
along z axis
F  [ 82  62 ]1 / 2  10
r F sin( )  50 sin( )  sin( )  1
 
o
   90 that is F  r
OR Use: |  |  | r F|  r F cos( )

F


r
 50 cos( )
 
r  F   (3x 8) î  î  (3x 6) î  ĵ  (4x 8) ĵ  î  (4x 6) ĵ  ĵ
  24  24  0
 
o
 50 cos()  0 so   90 that is F  r
Copyright R. Janow - Fall 2011
Field concepts - mathematical view
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A FIELD assigns a value to every point in space (2D, 3D, 4D)
It obeys some mathematical rules:
• E.g. superposition, continuity, smooth variation, multiplication,..
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A scalar field maps a vector into a scalar:
f: R3->R1
• Temperature, barometric pressure, potential energy
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A vector field maps a vector into a vector:
f: R3->R3
• Wind velocity, water velocity (flow), acceleration
• A vector quantity is assigned to every point in space
• Somewhat taxing to the imagination, involved to calculate
Example: map of the velocity
of westerly winds flowing
past mountains
FIELD LINES
FIELD LINES (streamlines)
show wind direction
Line spacing shows speed: dense  fast
Set scale by choosing how many lines to draw
Lines begin & end only on sources or sinks
Pick single
altitudes and make
slices to create
maps
Copyright R. Janow - Fall 2011
ISOBARS
EQUIPOTENTIALS
Scalar field examples
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A scalar field assigns a simple number
to be the field value at every point in
“space”, as in this temperature map.
•
Another scalar field: height at
points on a mountain. Contours
measure constant altitude
Contours far
apart
Contours
close
together
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Contours
Grade (or slope or gradient) is
related to the horizontal
spacing of contours (vector
field)
flatter
steeper
Copyright R. Janow - Fall 2011
Side View
Slope, Grade, Gradients (another field) and Gravity
Height contours h, can also portray potential energy U = mgh. If you move along a
contour, your height does not change, so your potential energy does not change. If
you move perpendicular to a contour, you are moving along the gradient.
•
Slope and grade mean the same
thing. A 6% grade is a slope of
lim h / x  dh / dx  0.06
6
x  0
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Gradient is measured along the path.
For the case above it would be:
100
lim h / l  dh / dl   6/100.2  0.06
x  0
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Gravitational force for example is the
gradient of potential energy
F  dU / dl  d (mgh) / dl  mg dh / dl
dl
dh / dl  sin()
dh
F  mg sin()
•
•

The steepness and/or force above are related to the GRADIENTS of height and/or
gravitational potential energy, respectively, and are also fields.
Are the GRADIENTS of scalar fields also scalar fields or are they vector fields?
Copyright R. Janow - Fall 2011
Vector Fields
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For a vector field the field value at every
point in space is a vector – that is, it has
both magnitude and direction
•
A vector field like the altitude gradient can
be defined by contours (e.g., lines of
constant potential energy – a scalar field).
The gradient field lines are perpendicular to
the altitude contours
•
The steeper the gradient (e.g., rate of
change of gravitational potential energy)
the larger the field magnitude is.
DIRECTION
•
The gradient vectors point along the
direction of steepest descent, which is also
perpendicular to the contours (lines of
constant potential energy).
•
Imagine rain on the mountain. The vectors
are also “streamlines.” Water running down
the mountain will follow these streamlines.
Copyright R. Janow - Fall 2011
Side View
Another scalar field – atmospheric pressure
Isobars: lines
of constant
pressure
How do the isobars affect air motion?
What is the black arrow showing?
Copyright R. Janow - Fall 2011
A related vector field: wind velocity
Copyright R. Janow - Fall 2011
Wind speed and
direction depend on
the pressure gradient
Visualizing Fields
Could be:
• 2 hills,
• 2 charges
• 2 masses
Scalar field: lines of constant field magnitude
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Altitude / topography – contour map
Pressure – isobars, temperature – isotherms
Potential energy (gravity, electric)
Vector field: field lines show a gradient
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•
Direction shown by TANGENT to field line
Magnitude shown by line density - distance
between lines
Lines start and end on sources and sinks
(highs and lows)
Forces are fields, but not quite what we call
gravitational, electric, or magnetic field
Mass or
negative
charge
Examples of scalar and vector fields in mechanics and E&M:
TYPE
MECHANICS
(GRAVITY)
FORCE
LAW
FORCE = SGMm / r2
SCALAR
FIELDS
GRAV POTENTIAL ENERGY,
GRAV POTENTIAL
(PE / UNIT MASS)
ag = FORCE / UNIT MASS
VECTOR
=“GRAV. FIELD”
FIELDS
of GRAVITY
Copyright =R.ACCELERATION
Janow - Fall 2011
ELECTROSTATICS
(CHARGE)
COULOMB FORCE
Magnetic field
around a wire
carrying current
MAGNETOSTATICS
(CURRENT)
MAG FORCE = q v X B
ELECTRIC POTENTIAL ENERGY
ELECTRIC POTENTIAL
(PE / UNIT CHARGE)
MAGNETIC P. E. OF A CURRENT
E = FORCE / UNIT CHARGE
= “ELECTRIC FIELD”
B = FORCE / CURRENT.LENGTH
= “MAGNETIC FIELD”
Fields are used to explain “Action at a Distance” (Newton)
• A test mass, test charge, or test current placed at some test point in a
field feels a force due to the presence of a remote source of field.
• The source “alters space” at every test point in its vicinity.
• The forces (vectors) at a test point due to multiple sources add up
via superposition (the individual field vectors add & cause the force).
Field Type
Definition
Source
Acts on
Strength
(dimensions)
gravitational
Force per unit mass
at test point
mass
another
mass
ag = F g / m
electrostatic
Force per unit
charge at test point
charge
another
charge
E=F/q
magnetic
Force per unit
current.length
electric
current
another
current
B
Copyright R. Janow - Fall 2011
Gravitation is a Vector Field
•
•
The force of Earth’s gravity
points everywhere in the
direction of the center of
the Earth.
The strength of the force is:

GMm
F   2 r̂
r
•
•
This is an inverse-square
force (proportional to the
inverse square of the
distance).
The force is a field
mathematically, but it is not
quite what we call
“gravitational field”.
Copyright R. Janow - Fall 2011
m
M
Idea of a test mass
•
•
The amount of force at some point
depends on the mass m at that point

GMm
F   2 r̂
r
What is the force per unit mass? Put a
unit test mass m near the Earth, and
observing the effect on it:

F
GM
  2 r̂  g(r )r̂
m
r
•
g(r) is the “gravitational field”, or
the gravitational acceleration.
•
The direction (only) is given by r̂
•
g(r) vector field, like the force.
Copyright R. Janow - Fall 2011
m
M
Same idea for
test charges & currents
Meaning of g(r):
1-2: What are the units of:
A.
B.
C.
D.
E.

F
GM
  2 r̂  g(r )r̂
m
r
?
Newtons/meter (N/m)
Meters per second squared (m/s2)
Newtons/kilogram (N/kg)
Both B and C
Furlongs/fortnight

F
GM
1-3: Can you suggest another name for
  2 r̂  g(r )r̂ ?
m
r
A.
B.
C.
D.
E.
Gravitational constant
Gravitational energy
Acceleration of gravity
Gravitational potential
Force of gravity
Copyright R. Janow - Fall 2011
Superposition of fields (gravitational)
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•
•
“Action-at-a-distance”: gravitational field permeates all of space with force/unit mass.
“Field lines” show the direction and strength of the field – move a “test mass” around to map it.
Field cannot be seen or touched and only affects the masses other than the one that created it.
•
What if we have several masses? Superposition—just vector sum the
individual fields.
M
•
M
M
M
The NET force vectors show the direction and strength of the NET field.
The same ideas apply to electric & magnetic fields
Copyright R. Janow - Fall 2011
Summarizing: Gravitational field of a point mass M
The gravitational field at a point is the acceleration of gravity g
(including direction) felt by a test mass at that point
• Move test mass m around to map direction
& strength of force
• Field g = force/unit test mass
• Lines show direction of g is radially inward
(means gravity is always attractive)
• g is large where lines are close together
inward force
on test mass m
gA
surfaces of
constant
field & PE
gB
rb
• Newton:

GM
g   2 r̂
r
rA
2
(Newtons/k g or m/s )
• Field lines END on masses (sources)
Where do gravitational field lines BEGIN?
• Gravitation is always attractive, lines BEGIN at r = infinity
Why inverse-square laws? Why not inverse cube, say?
Copyright R. Janow - Fall 2011
M
gB
gA
How long does it take for field to change?
Changes in field must propagate from source out to
observation point (test mass) at P.
 For Gravitation, gravity waves
 For electromagnetism, light waves
Action at a distance for E&M travels at speed of light
Copyright R. Janow - Fall 2011
An important idea called Flux (symbol F)
is basically a vector field magnitude x area
- fluid volume or mass flow
- gravitational - electric - magnetic
Definition: differential amount of flux dFg of field ag crossing vector area dA
ag
“unit normal”
n̂
outward and
perpendicular to
surface dA


dF g  flux of a g through dA

 a g  n̂dA (a scalar)
Flux through a closed or open surface S: calculate “surface integral” of field over S
Evaluate integrand at all points on
surface S
FS 
 dF 

a
 g  n̂dA
S
S
EXAMPLE : FLUX THROUGH A CLOSED EMPTY BOX IN A
UNIFORM g FIELD
• zero mass inside
• F from each side = 0 since a.n = 0, F from ends cancels
• TOTAL F = 0
• Example could also apply to fluid flow
n̂
n̂
n̂
ag
What if a mass (flux source) is in the box?
Can field be uniform? Can net flux be zero.
Copyright R. Janow - Fall 2011
n̂
FLUID FLUX EXAMPLE: WATER FLOWING ALONG A STREAM
Assume:
•
•
•
•
constant mass density
constant velocity parallel to banks
no turbulence (laminar flow)
incompressible fluid – constant r
n̂'
n̂

A 2
Flux measures the flow (current):
• rate of volume flow past a point
• rate of mass flow past point
• flows mean amount/unit time across area
2 related fields (currents/unit area):
• velocity v represents volume flow/unit area/unit time
• J = mass flow/unit area/unit time

A 1


J  rv

A  n̂A
n̂ is the outward unit vector
 to vector area A
Flux = amount of field crossing an area per unit time (field x area)


 

V  l  A 
 l  vt
The chunk of

volume flux 

 v  A
mass moves l in

t

t
A


time t:
mass of solid chunk  m  rV  r  l  A


 

v

m
l
 mass flux 
r
 A  rv  A  J  A
and
t
t
Continuity: net flux (fluid flow) through a closed surface = 0
………unless a source or drain is inside
Copyright R. Janow - Fall 2011
Gauss’ Law for gravitational field: The flux through a closed surface S
depends only on the enclosed mass (source of field), not on the details of S
or anything else

GM
Gravitational field: g   2 r̂
r
2
(Newtons/k g or m/s )
Consider two closed spherical shells,
radii rA & rB centered on M
inward force
on test mass m
surfaces of
constant
field & PE
Find flux through each closed surface
F A  gA xA A  
FB  gBxAB  
GM
rA2
GM
rB2
  4GM
gA
x 4rB2   4GM
r
x
4rA2
Same! – Flux depends only on
the enclosed mass
gB
rb
M
gB
A
FLUX measures the source strength inside of a closed
surface - “GAUSS’ LAW”
Copyright R. Janow - Fall 2011
gA
The Shell Theorem follows from Gauss’s Law
1. The force (field) on a test particle OUTSIDE a uniform SPHERICAL shell
of mass is the same as that due to a point mass concentrated at the
shell’s mass center (use Gauss’ Law & symmetry)
m
r
x
r
m
x
Same for a solid sphere (e.g., Earth, Sun) via nested shells
m
r
r
x
x
+
r
x
+
2. For a test mass INSIDE a uniform SPHERICAL shell of mass m,
the shell’s gravitational force (field) is zero
• Obvious by symmetry for center point
• Elsewhere, integrate over sphere (painful) or
apply Gauss’ Law & Symmetry
m
x x
3. Inside a solid sphere combine the above. The force on a test mass
INSIDE depends only on mass closer to the CM than the test mass.
x
• Example: On surface, measure acceleration
distance
r
from center
• Halfway to center,
Copyright R. Janow - Fall 2011
ag = g/2
g
a
4
3
Vsphere  r 3
When you are solving physics problems, two ways to
approach problem
 Brute force….. Solve equations in 3-D geometry
 Use intuition to wisely choose a coordinate
system and symmetry which help you.
How do you choose
coordinate system
to simplify
problem?
What direction is x
and y direction?
Copyright R. Janow - Fall 2011
Symmetry
Spherical Symmetry
Use Spherical
Coordinates
Copyright R. Janow - Fall 2011
Symmetry
Eg. current in a long,
Straight wire
Cylindrical Symmetry
Use Cylindrical
(polar) Coordinates
End on view
Copyright R. Janow - Fall 2011
Symmetry
Planar Symmetry
Straight
Field lines
Use Cartesian
Coordinates
Copyright R. Janow - Fall 2011
Curved
Field lines? WHY?
Example: Calculate the field (gravitation) due to
a simple source (mass distribution) using superposition
Find the field at point P on x-axis due to
two identical mass chunks m at +/- y0
y
m
• Superposition says add fields created at P
by each mass chunk (as vectors!!)
• Same distances r to P for both masses
r0
+y0
r02  x 02  y02

+x0
• Same angles with x-axis
cos()  x 0 / r0
•
Gm
x 02  y02

ag
-y0
r0
• Same magnitude ag for each field vector
ag 
ag
(from Newtons law of gravitatio n)
m
y components of fields at P cancel, x-components reinforce each other
a tot  a x  
2Gm cos()
r02
 
2Gm x 0
r03
where
r03  [ x 02  y02 ]3 / 2
• Result simplified because problem has a lot of symmetry
Direction: negative x
Copyright R. Janow - Fall 2011
P
x
Example: Calculate gravitational field due to mass
distributed uniformly along an infinitely long line
Find the field at point P on x-axis
to y  
• Similar approach to previous example, but need to
include mass from y = – infinity to y =+ infinity
• Superposition again:
dm = ldy
add differential amounts of field created at P by
differential mass chunks at y (as vectors!!)
r
dag
y
• As before, y components of fields cancel, xcomponents reinforce each other for symmetrically

located chunks
x

P
Gdm
-y
da x  dag cos()   2 cos()
r
l = mass/unit length
• Mass per unit length l is uniform, find dm in terms of :
x
dm  ldy  lx[1  tan2 ()] d
 da x  
Glx[1  tan2 ()]
x 2 [1  tan2 ()]
cos() d  
Gl
cos() d
x
• Integrate over  from –/2 to +/2
 ax  
Gl   / 2
Gl
cos(

)
d



2
x   / 2
x
Field of an infinite line falls off as 1/x not 1/x2
Copyright R. Janow - Fall 2011
to y
 
y  x tan()
dy
dtan()
x
 x [1  tan2 ()]
d
d
r 2  x 2  y2  x 2 [1  tan2 ()]
 /2
 cos()d  2
- /2
Line integral (path integral) examples
for a gravitational field
How much work is done on a test mass
 as it traverses a particular path


dW  dU  F  ds  mag  ds
through a field?
B

 U    F  ds
evaluate along path
A
Gravitational field is conservative so U is
independent of path chosen
B
A


F

d
s

F

d
s


for any path between A & B
A
 B
U   F  ds  0 for any path closed chosen
S
circulation,or path integral
EXAMPLE
uniform field
U= - mgh
Copyright R. Janow - Fall 2011
U= + mgh
test mass
Spherical Polar Coordinates in 3 Dimensions (Extra)
+z
Cartesian

r  (x, y, z)

r  x î  yĵ  zk̂

r
z

Polar, 3D r  (r, , )
rz  r cos()

  " colatitude" , in [0,  ] radians
  " azimuth" , in [0,2  ] radians
r  (x 2  y 2  z 2 )1 / 2
+y
x
Polar to Cartesian
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
| r |  rxy  r sin( )

90o
90o
z  r cos()
x  rxy cos()  r sin() cos()
y  rxy sin()  r sin() sin()
r2  x 2  y 2  r 2 sin 2 ()
P
90o
y
+x
Cartesian to Polar
  cos1 (z / r )
  tan-1(y / x)
r  (x 2  y 2  z2 )1/ 2
Show that the surface area of a sphere = 4R2
(Advanced)
READ
Copyright R. Janow - Fall 2011
Gravitational field due to an infinite sheet of mass (Advanced)
Copyright R. Janow - Fall 2011
Does not depend
on distance from plane!