copyright r. janow - fall 2011 1 electricity and magnetism introduction to physics 121 syllabus,...

Copyright R. Janow - Fall 2011

1

Electricity and Magnetism Introduction to Physics 121

• Syllabus, rules, assignments, exams, etc.• iClickers• Quest• Course content overview

• Review of vector operations• Dot product, cross product

• 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


Course Content

• 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 endsAbout 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.


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

Phys 111 Phys 121 Phys 234

Mechanics Electromagnetics “Modern” Physics

Typically ECE and Physics majors



Capacitors, Resistors, inductors, and Kirchkoff’s loop laws for circuits from Phys 121 are the basics for

• Computers



Electric fields, voltages, charges from Phys 121 are the basics for medicine and Biology

• Electro-cardiography• biological function of Cells



Electric fields, voltages, charges from Phys 121 are the basics for Civil Engineering infrastructure

• Power Grid• Sensors



Electric fields, voltages, charges, circuits from Phys 121 are the basics for Electrical Engineering

WiFi3G, 4G, 5G…….

lasers


Electricity and MagnetismLecture 01 - Vectors and Fields Physics 121

Review of Vectors :• Components in 2D & 3D. Addition & subtraction• Scalar multiplication, Dot product, vector productField 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


Vector Definitions

Representations in 2 Dimensions:

• Cartesian (x,y) coordinates

jA iA A yx

• magnitude & direction

22yx A A A

x

y1-

A

Aant

- Experiments tell us which physical quantities are scalars and vectors - E&M uses vectors for fields, vector products for magnetic field and force

• Addition and subtraction of vectors:

yyy andxxx means BAC BAC B A C

yy andxx means AC AC A- C

y

x

A

A

Ax = A cos()

Ay = A sin()

k

j

i

z

• Notation for vectors:

amF

amF

aF

m


Vectors in 3 dimensions

• Unit vector (Cartesian) notation:

• Spherical polar coordinate representation:

1 magnitude and 2 directions

• Conversion into x, y, z components

• Conversion from x, y, z components

) , ,a( a

cosaasinsinaacossinaa

z

yx

xy

z

zyx

a/atana/acos

aaaa

1

1

222

kajaiaa zyx

y

x

a

z

az

ayax

Rene Descartes 1596 - 1650


Definition: Right-Handed Coordinate Systems

• We always use right-handed coordinate systems.

• In three-dimensions the right-hand 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.

y

x

z

This course will use many right hand rules related to this one!


Right Handed Coordinate Systems

1-1: Which of these coordinate systems are right-handed?

A. I and II.B. II and III.C. I, II, and III.D. I and IV.E. IV only.

z

x

y II.

y

z

xIII.

z

y

x IV.

x

z

y I.


Vector Multiplication

jsAisA As yx

Multiplication by a scalar: A

As

vector times scalar vector whose length is multiplied by the scalar

Dot product (or Scalar product or Inner product):

y y z z ABcos( ) B A A B A B x xA B A B

- vector times vector scalar - projection of A on B or B on A - commutative

A

B

ˆ ˆ ˆ ˆˆ ˆ 0, j k 0, i k 0

ˆ ˆ ˆ ˆ ˆ ˆi i 1, j j 1, k k 1

i j


Vector multiplication, continued

Cross product (or Vector product or Outer product):

A

B

C

-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

e)commutativ(not A B - B A C

B to A from angle smaller the is where :magnitude

) ABsin( 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)

Algebra:)B(sA B)A(s BAs :rules eassociativ

CA BA )CB( A :rule eistributivd

)CB(A C)BA(

k)BA-B(A j)BA-B(A i)BA-B(A

)kB jBi(B )k A jAiA( B A

xyyxzxxzyzzy

zyxzyx

0kk 0,jj 0,ii

j- ki ,i kj ,k ji

Unit vector representation:

i

kj

Fr

prL

Bvq EqF

Applications:


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.

along z axis ˆ N.m k ˆ 5050

a) Fr

k k jj)( ij)( ji)( ii)(

)j6 i()j4 i(Fr

xxxx

321864846383

83

Use:

r

F

Use: )sin( F r ||

b)10 ]6 8[ F 5 ]4 3[ r / 22/ 22 2121

1 )sin( )sin( 50 )sin( F r

r F 90 isthat o

| | | r F| r F cos( ) 50 cos( ) OR Use:

0 jj)( ij)( ji)( ii)( Fr xxxx

242464846383

r F 90 0 )cos( 50 isthat o

so


Field concepts - mathematical view

• 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,..

• A scalar field maps a vector into a scalar: f: R3->R1

• Temperature, barometric pressure, potential energy

• 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

ISOBARSEQUIPOTENTIALS

FIELD LINES

Example: map of the velocity of westerly winds flowing past mountains

Pick single altitudes and make slices to create maps

FIELD LINES (streamlines) show wind directionLine spacing shows speed: dense fastSet scale by choosing how many lines to drawLines begin & end only on sources or sinks


Scalar field examples• 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

Side View

steeper

flatter

Contours

Contours close

together

Contours far apart

• Grade (or slope or gradient) is related to the horizontal spacing of contours (vector field)


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.

100

6

• 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?

0600

.dx/dhx/hlimx

• Slope and grade mean the same thing. A 6% grade is a slope of

)sin(mgF

)sin(dl/dh

dl/dh mgdl/)mgh( ddl/dUF

• Gradient is measured along the path. For the case above it would be:

dh

dl

• Gravitational force for example is the gradient of potential energy

06060

. /100.2 dl/dhl/hlimx


Vector Fields

• 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.

Side View


Another scalar field – atmospheric pressure

How do the isobars affect air motion? What is the black arrow showing?

Isobars: linesof constant pressure


A related vector field: wind velocity

Wind speed and direction depend on the pressure gradient


Visualizing Fields

Scalar field: lines of constant field magnitude• Altitude / topography – contour map• Pressure – isobars, temperature – isotherms• Potential energy (gravity, electric)

Vector field: field lines show a gradient • 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

Examples of scalar and vector fields in mechanics and E&M:

TYPE MECHANICS

(GRAVITY)

ELECTROSTATICS

(CHARGE)

MAGNETOSTATICS

(CURRENT)

FORCE LAW FORCE = GMm / r2 COULOMB FORCE MAG FORCE = q v X B

SCALAR

FIELDS

GRAV POTENTIAL ENERGY,

GRAV POTENTIAL

(PE / UNIT MASS)

ELECTRIC POTENTIAL ENERGY

ELECTRIC POTENTIAL

(PE / UNIT CHARGE)

MAGNETIC P. E. OF A CURRENT

VECTOR

FIELDS

ag = FORCE / UNIT MASS

=“GRAV. FIELD”

= ACCELERATION of GRAVITY

E = FORCE / UNIT CHARGE

= “ELECTRIC FIELD”

B = FORCE / CURRENT.LENGTH

= “MAGNETIC FIELD”

Could be:• 2 hills, • 2 charges• 2 masses

Mass or negative charge

Magnetic field around a wire

carrying current


Fields are used to explain “Action at a Distance” (Newton)

Field Type Definition(dimensions)

Source Acts on Strength

gravitationalForce per unit mass

at test pointmass

anothermass

ag = Fg / m

electrostaticForce per unit

charge at test pointcharge

another charge

E = F / q

magneticForce per unit current.length

electric current

another current

B

• 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).


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:

• 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”.

rr

GMmF

2

M

m


Idea of a test mass• The amount of force at some point

depends on the mass m at that point

• What is the force per unit mass? Put a unit test mass m near the Earth, and observing the effect on it:

• g(r) is the “gravitational field”, or the gravitational acceleration.

• The direction (only) is given by

• g(r) vector field, like the force.

rr

GMmF

2

r)r(grr

GM

m

F

2

r

M

Same idea for test charges & currents

m


Meaning of g(r):

1-2: What are the units of: ?

A. Newtons/meter (N/m)B. Meters per second squared (m/s2)C. Newtons/kilogram (N/kg)D. Both B and CE. Furlongs/fortnight

r)r(grr

GM

m

F

2

1-3: Can you suggest another name for ?

A. Gravitational constantB. Gravitational energyC. Acceleration of gravityD. Gravitational potentialE. Force of gravity

r)r(grr

GM

m

F

2


Superposition of fields (gravitational)• “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.

The same ideas apply to electric & magnetic fields

M MM M

• The NET force vectors show the direction and strength of the NET field.


The gravitational field at a point is the acceleration of gravity g

(including direction) felt by a test mass at that point

Summarizing: Gravitational field of a point mass M

M

rb

rA

gAgB

gA

gB

surfaces ofconstant field & PE

inward forceon test mass m• 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

• Newton:

)m/s or g(Newtons/k2 r

r

GMg

2

Where do gravitational field lines BEGIN?• Gravitation is always attractive, lines BEGIN at r = infinity Why inverse-square laws? Why not inverse cube, say?

• Field lines END on masses (sources)


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


An important idea called Flux (symbol is basically a vector field magnitude x area

Definition: differential amount of flux dg of field ag crossing vector area dA

scalar) (adA n a Ad through a of flux d

g

gg

ag n

“unit normal”

outward andperpendicular to

surface dA

- fluid volume or mass flow - gravitational - electric - magnetic

Flux through a closed or open surface S: calculate “surface integral” of field over S

dA n a d S S

gS

Evaluate integrand at all points on surface S

EXAMPLE : FLUX THROUGH A CLOSED EMPTY BOX IN A UNIFORM g FIELD• zero mass inside• from each side = 0 since a.n = 0, from ends cancels• TOTAL = 0• Example could also apply to fluid flow ag

n

n

n

n

What if a mass (flux source) is in the box? Can field be uniform? Can net flux be zero.


FLUID FLUX EXAMPLE: WATER FLOWING ALONG A STREAMAssume: • constant mass density• constant velocity parallel to banks• no turbulence (laminar flow)• incompressible fluid – constant

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 v J

Flux = amount of field crossing an area per unit time (field x area)

Av t

Al

t

V flux olumev

and A J Av A t

l

t

m luxf mass

A l V m chunk solid of mass

The chunk of mass moves l in time t:

v

tvl

A

Continuity: net flux (fluid flow) through a closed surface = 0 ………unless a source or drain is inside

A n

AnA

area vector to

vectorunit outward the is

1A

2A

'n

n


M

rb

rAgAgB

gA

gB

surfaces ofconstant field & PE

inward forceon test mass m

GM r4 x r

GMxAg 2

BB

BBB 42

Consider two closed spherical shells, radii rA & rB centered on M

Find flux through each closed surface GM r4 x

r

GMxAg 2

AA

AAA 42

)m/s or g(Newtons/k2 r

r

GMg

2

Gravitational field:

Same! – Flux depends only on the enclosed mass

FLUX measures the source strength inside of a closed surface - “GAUSS’ LAW”

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


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)

x

mr

xm r

Same for a solid sphere (e.g., Earth, Sun) via nested shells

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

x

mx

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 g a

distance r from center

• Halfway to center, ag = g/2

33

4 rVsphere

xr

x

mr x

r

+ +


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?


Symmetry

Spherical Symmetry

Use Spherical Coordinates


Symmetry

Cylindrical Symmetry

Use Cylindrical (polar) Coordinates

Eg. current in a long, Straight wire

End on view


Symmetry

Planar Symmetry

Use Cartesian Coordinates

StraightField lines

CurvedField 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

• Superposition says add fields created at P by each mass chunk (as vectors!!)• Same distances r to P for both masses

• Same angles with x-axis

• Same magnitude ag for each field vector

• y components of fields at P cancel, x-components reinforce each other

• Result simplified because problem has a lot of symmetry

y

xP

m

m

+y0

-y0

+x0

ag

ag

r0

r0

20

20

20 y x r

00 r / x )cos(

n)gravitatio oflaw Newtons (fromg yx

mG a

20

20

2320

20

303

0

020

/ wherextot ]yx[ r

r

x m2G

r

)cos( m2G a a

Direction: negative x


Example: Calculate gravitational field due to mass distributed uniformly along an infinitely long line Find the field at point P on x-axis• Similar approach to previous example, but need to

include mass from y = – infinity to y =+ infinity• Superposition again:

add differential amounts of field created at P by

differential mass chunks at y (as vectors!!)• As before, y components of fields cancel, x-

components reinforce each other for symmetrically located chunks

• Mass per unit length is uniform, find dm in terms of :

• Integrate over from –/2 to +/2 )tan( x y

)](tan[ xy x r 22222 1

)](tan[1 xd

)dtan(x

d

dy 2

)cos( r

mdG )cos(da da gx

2

d )](tanx[1 dy dm 2

d )cos(x

G d )cos(

)](tan[1x

)](tanx[1G da

2

2

x 2

x

G d )cos(

x

G a

/

/x

2

2

2

Field of an infinite line falls off as 1/x not 1/x2 2 )dcos(

/2

/2-

xP

dm = dy

y

x

dagr

y to

y to

= mass/unit length

-y


Line integral (path integral) examples for a gravitational field

How much work is done on a test mass as it traverses a particular path

through a field? sdmasdFdUdW g

path along evaluate

B

A

sdF U

test mass

Gravitational field is conservative so U is independent of path chosen

B & A between path any for

A

B

B

A

sdF- sdF

chosen closed path any for

S

sdF U 0

circulation,or path integral

EXAMPLEuniform field

U= - mgh U= + mgh


Spherical Polar Coordinates in 3 Dimensions (Extra)

+x

+y

+z

P

r

)sin(r r |r| xy

)cos(rrz

90o90o

y

x

90o

zkz j y ix r

z)y,(x, r

),(r, r

21 /222radians][0,2in ,azimuth""

radians ][0,in ,olatitude"c"

)z y (x r

Cartesian

Polar, 3D

)(sinr yx r

)sin()sin(r )sin(r y222

xy

22

)cos()sin(r )cos(r x xy )cos(r z

Polar to Cartesian

)x/y(tan -1

)r/z(cos 1

21 /222 )z y (x r

Cartesian to Polar


READShow that the surface area of a sphere = 4R2

(Advanced)


Gravitational field due to an infinite sheet of mass (Advanced)

Does not dependon distance from plane!

copyright r. janow - fall 2011 1 electricity and magnetism introduction to physics 121 syllabus,...

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