Download Review for final

Review for final
Schedule
• Dec. 3-5 (Wed-Fri) – Review for final
! Dec.
3 (Wed) – HW set #12 due at 11pm
! Dec.
8 (Mon) – Corrections #3 due at 7am
! Dec.
8 (Mon) – Final Exam 5:45-7:45pm
• Section 1 – N130 BCC (Business College)
• Section 2 – 158 NR (Natural Resources)
Final Exam
• If >2 finals on Mon. may take make-up final
exam on Tues. from 8-10am
!
Email Prof. Tollefson with list of other finals
• Allowed 3 sheets of notes (both sides)
and calculator
• Covers Chapters 22-37, HW sets 1-12
• Exam will have 20 questions
• Need photo ID
Electric Fields
• Point change q
F =k
!
!
q q0
2
q
E=k 2
r
q
V =k
r
r
E points away from positive
charges and towards negative
charges
Superposition principle (many
charges)
r r r
r
F = F1 + F2 + ... + Fn
r r r
r
E = E1 + E2 + ... + En
n
V = ∑Vi
i =1
Electric Potential
!
Electric potential from field
r
r
∆V = − ∫ E • ds
f
i
!
Constant field over distance d
∆ V = − Ed
!
Work done moving charge q0
W = q0 ∆V
Electric Potential
!
Blue lines are the electric field
!
Dashed lines are equipotential
surfaces where all points are
at the same potential
!
V decreases by constant
intervals from the positive
charge to the negative
charge
Guass’ Law
!
Electric flux
r r
Φ = ∫ E • dA
r
r
E • dA = E dA cos θ
!
Gauss’ Law
!
Net charge qenc is sum of all enclosed
charges and may be +, -, or zero
r r
ε 0 Φ = ε 0 ∫ E • dA = qenc
Conductors (example)
!
!
!
!
!
!
Charge q1 inside
E=0 inside conductor
Thus Φ=0 for Gaussian
surface (red line)
So net charge enclosed
must be 0
Induced charge of
q2 = -q1 lies on inner
wall of conductor
Shell is neutral so
charge of q3 = -q2 on
outer wall
Electric Potential Energy
!
Work required to assemble
the charges
11
12
U =U12 +U13 +U14
+U23 +U24 +U34
!
where
etc
q1q2
U12 = k
d
13
14
Motion in a B Field
!
Force on a charged particle due to
a magnetic field is
!
FB does not change the speed
(magnitude of v ) or kinetic energy
r
r r
FB = qv × B
of particle
!
Charged particles moving with v ⊥
to a B field move in a circular path
with radius, r
!
Force on a current carrying wire
due to a magnetic field is
mv
r=
qB
r
r r
FB = iL × B
Motion in a B Field
r
r r
FB = qv × B
• Right-hand rule – For
positive charges - when
the fingers sweep v into
B through the smaller
angle φ the thumb will
be pointing in the
direction of FB
!
For negative charges FB
points in opposite
direction
Motion in a B Field
mv
r=
qB
!
Circular motion
!
Period (time for one
revolution)
2π r 2π m
=
T=
v
qB
!
Frequency (the number of
revolutions per unit time)
!
Angular frequency:
1
f =
T
ω = 2π f
B Fields from Currents
!
B field a distance R from a long
straight wire carrying current i
µ 0i
B=
2π R
!
B field is tangent to magnetic field
lines
B Fields from Currents
!
right-hand rule
!
Point thumb in direction
of current flow
!
Fingers will curl in the
direction of the magnetic
field lines due to current
B Fields from Currents
!
Force on a wire carrying current, i1, due to B of
another parallel wire with current i2
µ 0 Li1i2
F=
2πd
!
!
Force is attractive if current in both wires are in the
same directions
Force is repulsive if current in both wires are in the
opposite directions
B Fields from Currents
!
For ideal solenoid:
B = µ0in
n is the number of turns/length
Faraday’s Law
!
Magnetic flux
r r
Φ B = ∫ B • dA = BA cosθ
!
Faraday’s law (N loops)
!
Lenz’s law – induced emf
gives rise to a current
whose B field opposes
the change in flux that
produced it
dΦ B
E = −N
dt
Faraday’s law
• We can change the magnetic flux
through a loop (or coil) by:
!
Changing magnitude of B field within
coil
!
Changing area of coil, or portion of
area within B field
!
Changing angle between B field and
area of coil (e.g. rotating coil)
dB
E = − NAcosθ
dt
dA
E = − NB cosθ
dt
d( cosθ )
E = − NBA
dt
Generators
!
!
Generator with N turns of
area A and rotating with
constant angular velocity,
ω
Magnetic flux is
Φ B = BA cos ω t
!
Emf is
E = NBAω sinωt
t
Circuits
!
Current
!
Resistors
!
!
dq
i=
dt
Ohm’s law
Power lost
V = iR
2
V
P = iV = i R =
R
2
Circuits
!
Definition of capacitance: C = q
V
!
Parallel plates of area A and separation d
C=
ε0 A
d
Circuits
!
Junction rule:
iin = iout
!
Loop rule: sum of
potential changes around
each loop is zero
Circuits
!
Parallel
C eq =
Capacitors
Series
n
∑C
i
i
V same on each
! Total q is sum
!
!
Resistors
1
=
R eq
!
n
∑
i
1
Ri
V same on each
1
=
C eq
n
∑
i
1
Ci
q same on each
! Total V is sum
!
n
Req = ∑ Ri
i
!
i same in each
!
Total V is sum
Elements of RLC circuits
Resistor
!
!
Energy
stored
Voltage
change
U
= 0
V = (− )iR
Inductor
U
B
1
= Li
2
di
V = −L
dt
Capacitor
2
U
E
1 q2
=
2 C
q
V = (− )
C
(-) means sign relative to the direction of current flow
!
Power lost
P=i R
2
P=0
P=0
LC Circuits
!
Charge
q = Q cos(ωt )
dq
i=
= −Qω sin( ωt )
dt
!
Current
!
Angular frequency
!
No power loss
ω=
1
LC
RLC Circuits
!
Charge on capacitor
q = Qe
!
− Rt / 2 L
cos(ω ′t )
Angular frequency
ω ′ = ω 2 − ( R / 2 L)2
!
Natural frequency
ω=
1
LC
AC Circuits
Resistive load
E = E msinω d t , ω d = driving frequency
VR
IR =
R
Capacitive load
VC
IC =
XC
1
XC =
ωd C
Inductive load
VL
IL =
XL
X L = ωd L
X is the reactance
AC Circuits
E = E msinω d t , ω d = driving frequency
!
Input
!
Current (same everywhere)
Em
I=
Z
i = I sin(ωd t − φ )
X L − XC
tan φ =
R
!
Solution
!
Z is the impedance
!
I is maximum on resonance where
X L = XC
X L ( ωd )
=
XC
ω2
Z = R 2 + ( X L − X C )2
Z=R
ωd = ω
2
EM Waves
E = E m sin( kx − ω t )
B = Bm sin( kx − ω t )
v = c = fλ =
1
µ 0ε 0
= 3 × 108 m/s
Direction and power per unit area
Intensity
I=
1
2µ 0c
E m2
r 1 r r
S=
E×B
Pressure
µ0
I
pr =
c
2I
pr =
c
absorption reflection
Polarization
!
Polarization is the direction of the E
field
!
Intensity of unpolarized light with
intensity I0 after hitting a polarizing
sheet
I =
!
Intensity of polarized light with
intensity I0 after hitting a polarizing
sheet
I = I 0 cos θ
1
2
I0
2
Reflection & Refraction
!
Reflection: θ 1′ = θ 1
!
Refraction (Snell’s law)
n2 sin θ 2 = n1 sin θ1
• Index of refraction
speed in vacuum
c
n=
=
speed in medium
v
ω1 = ω 2
• Critical angle (no refracted wave)
v1 λ1 n2
=
=
v2 λ2 n1
n2
θ C = sin
n1
−1
Mirrors
!
!
!
Plane – flat mirror
Concave – caved in towards object
Convex – flexed out away from object
1 1
1
+ =
p i
f
!
!
!
!
!
!
!
f = 12 r
r = radius of curvature
f = focal length, f>0 concave, f<0 convex
p = position of object
i = position of image
real images on side where object is
virtual images on opposite side
lateral magnification:
h′
m =
h
i
m=−
p
Thin Lenses
!
!
!
!
!
Real images: opposite side - virtual images: same side
Diverging lens (f<0): smaller, same orientation, virtual images
Converging lens (f>0): both real and virtual images
Image position and magnification: 1
1
1
Lens maker’s equation:
1 1
1
= (n − 1) − 
f
 r1 r2 
p
+
i
=
f
i
m=−
p
Interference and Diffraction
• Diffraction minima given by
(a=slit width)
y' =
a sinθ ' = m' λ
m' λ D
, m' = 1,2...
a
Two-slit maxima given by
(d=slit separation) d sin θ =
!
mλ
mλ D
y=
, m = 0,1,2...
d
!
Small angles
sinθ = θ (θ in radians)
Thin Films
!
Phase change at interface
! Refraction at interface (the transmitted wave) never
changes phase
! Reflection gives a phase change
1
2
!
λ if n1 < n2
Phase change due to path a-b-c in
material n2 over a total length 2L
!
1 and 2 are in phase if
2 L = mλn 2 , m = 0,1,2...
!
1 and 2 are out of phase if
2 L = ( m + ) λn 2 , m = 0,1,2...
1
2
λn 2 =
λ
n2