Download Formula Sheet for Exam #2

Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
Formula Sheet for Exam #2
Reading and thoroughly familiarizing yourself with this formula sheet is an important part
of, but it is not a substitute for, proper exam preparation. The latter requires, among other
things, that you have re-worked all assigned homework problem sets (PS) and the in-class
quizzes, studied the posted PS solutions, and worked and studied the assigned conceptual
practice (CP) problems, as well as (optionally) some practice test (PT) problems, as posted
on the LON-CAPA homework and on the PHYS1112 examples and homework web pages.
You should consult the syllabus, and in particular review the Class Schedule on the last
syllabus page (posted on the PYS1112 course web site), to find out which topics you should
cover in preparing for this exam.
Wave Optics, Interference, Diffraction
(1) Periodic Wave Condition:
v = λf =
λ
τ
(2) Index of Refraction for electromagnetic waves, definition:
n=
c
λvacuum
=
v
λ
with λvacuum ≡ c/f = cτ .
(3) Definition of Path Length Difference for two-source, double-slit or adjacent slits in
multi-slit/diffraction grating:
∆` ≡ `2 − `1
(4) Path Length Difference vs. Angle: ∆` is given approximately in terms of observation
angle θ measured from central axis:
∆` ∼
= d sin θ
if
dL
where L =distance from slits or sources to observation screen, d =spacing of adjacent sources,
slits or lines in double-slit, multi-slit or diffraction grating.
(5) Constructive Interference Condition (≡ intensity maxima, principal maxima, bright
fringes) for two-source, double-slit, multi-slit or diffraction grating:
∆` = mλ
or
d L);
d sin θ = mλ (if
with m = 0, ±1, ±2, ...
where m is the ”order” of the (principal) maximum.
(6) Destructive Interference Condition 1 (≡ intensity minima, dark fringes) for twosource or double-slit experiment:
∆` = m +
1
λ
2
or
d sin θ = m +
1
λ (if d L);
2
1
with
m+
1
1 3
= ± , ± , ...
2
2 2
Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
(7) Destructive Interference Condition 2 (≡ intensity minima, dark fringes) for singleslit diffration with W =slit width and W L:
W sin θ = mλ
with m = ±1, ±2, ... (but m 6= 0).
(8) Destructive Interference Condition 3 for first intensity minimum of circular aperture
diffration with W =aperture diameter and W L:
W sin θ ∼
= 1.22λ (for 1st circular intensity minimum).
Charge, Electric Force, Electric Field, Flux, Gauss’s Law
(1) Coulomb’s Law for force F ≡ |F~ | on two point charges Q1 and Q2 at distance r:
F =k
|Q1 ||Q2 |
r2
with k = 8.99 × 109 Nm2 /C2 .
~ in terms of electric force F~ exerted on test charge q
(2) Definition of Electric Field E
~ generated by ”other” charges):
(with E
~
~ ≡F ;
E
q
~ .
hence F~ = q E
~
~ depends
Note E-field
is independent of the test charge q used to detect the electric force F~ : E
~
only on the ”other” charges which generate E.
~ by Point Charge Q, at an observation point P with distance
(3) Electric Field E ≡ |E|
r from Q:
|Q|
E=k 2
r
~ pointing radially away from positive charge, Q > 0; or radially towards negative
with E
charge, Q < 0.
~ by Uniform Surface Charge Density σ ≡ Q/A on a single
(4) Electric Field E ≡ |E|
planar surface of area A with total charge Q, in close proximity to the surface:
E=
|σ|
|Q|
=
2o
2o A
~ ∼
~
with E
= const (uniform E−field)
~ normal to the surface, pointing away
where o ≡ 1/(4πk) = 8.85 × 10−12 C2 /Nm2 ; and E
from positively charged (Q > 0) or towards negatively charged (Q < 0) surface.
~ in Planar Capacitor: between closely spaced, parallel, planar
(5) Electric field E ≡ |E|
plates of charges Q and −Q, and opposing surface areas A, without dielectric (κ = 1):
E=
|Q|
o A
~ ∼
~
with E
= const (uniform E−field)
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Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
~ normal to the plate surfaces, pointing from positive towards negative plate.
with E
~ is being generated
(6) Superposition Principle of Electric Field: If an electric field E
~
by multiple charged objects (Q1 , Q2 , ...), then E at any observation point P is the vector
~ 1, E
~ 2 , ... that would be generated
sum (resultant vector) of the electric field contributions E
by each of the charged objects in isolation at that point P :
~ =E
~1 + E
~ 2 + ...
E
~
(7) Electric Flux Φ of constant E-field
through planar surface of area A with the surface
o
~ with 0 ≤ θ ≤ 180o :
normal at angle θ from E,
Φ = EA cos θ
~
(8) Gauss’s Law, relating total electric flux Φ(S) of the E-field
through a closed surface S
(with outward-directed surface normal) to the total charge Q(S) enclosed inside S:
Φ(S) =
1
Q(S)
o
Electric Potential, Potential Energy, Capacitance, Electric Energy Storage
(1) Definition of Electric Potential V and electric potential difference ∆V (also known
as ”voltage drop”), in terms of potential energy U and potential energy difference ∆U ,
~
respectively, for a test charge q moving or being moved through E-field:
V = U/q ,
∆V = ∆U/q ;
hence U = qV ,
∆U = q∆V .
~
~
Note that V or ∆V is a property of the E-field,
depends only on the E-field,
and is
therefore independent of the test charge q.
~ = const):
(2) Electric Potential Difference in a Uniform Electric Field (E
∆V = −E ∆s cos θ
where ∆V ≡ VB − VA is the electric potential difference between points B and A; the vector
~ with
∆~s points from A to B with length ∆s ≡ |∆~s|; and θ is the angle between ∆~s and E
o
o
o
~ (θ < 90 ); and
0 ≤ θ ≤ 180 . Hence, ∆V < 0 when moving from A to B in direction of E
o
~ (θ > 90 ).
∆V > 0 when moving from A to B against direction of E
(3) Electric Potential for Point Charge Electric Field (with E = k|Q|/r2 ):
V =k
Q
r
where r is the distance from point charge Q to observation point.
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Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
~ is being gener(4) Superposition Principle of Electric Potential: If an electric field E
ated by multiple charged objects (Q1 , Q2 , ...), then its electric potential V at any observation
point P is the scalar sum (sum of numbers) of the electric potential contributions V1 , V2 ,
... that would be generated by each of the charged objects in isolation at that point P ; and
likewise for the electric potential difference ∆V ≡ VB − VA between any points A and B:
V = V1 + V2 + ...
or
∆V = ∆V1 + ∆V2 + ...
(5) Definition of Capacitance: For two oppositely charged metallic objects a and b,
with −Q stored on a and Q stored on b, their electric potential difference V ≡ Vb − Va is
proportional to the charge Q. The capacitance of the two metallic objects is then defined
as:
Q
Q
C≡
,
hence
Q = CV
or
V =
V
C
(6) Voltage and Capacitance of a Planar Capacitor: For two oppositely charged,
parallel planar metallic plates, each of opposing surface area A, closely spaced with distance
d, the voltage V and capacitance C are
|V | = Ed =
|Q|d
;
κo A
C≡
|Q|
A
= κo ≡ κ Co
|V |
d
where κ is the dielectric constant of the dielectric (insulating) material between the plates
and κ = 1 for vacuum or air, and Co ≡ o A/d is the capacitance without dielectric.
(7) Electric Field Energy Storage in a Capacitor: The energy UE required to build up
a charge Q and a voltage V = Q/C in a capacitor is stored as electric field energy between
the capacitor plates and it is given by
UE =
1 2 1
Q = CV 2
2C
2
Electric Field Energy Density. Energy per volume, uE , stored in an electric field is given
in terms of the field strength E
o
uE = E 2
2
Mechanics Memories: Velocity, Acceleration, Force, Energy, Power
(1) Velocity
~v =
∆~r
∆t
if constant; else ~v = lim
∆~r
∆t→0 ∆t
~a =
∆~v
∆t
if constant; else ~a = lim
(2) Acceleration
∆~v
∆t→0 ∆t
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Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
(3) Constant-Acceleration Linear Motion: for ∆~r ≡ ~rf − ~ri and ∆~v ≡ ~vf − ~vi
∆~r =
1
(~vi + ~vf ) t ;
2
∆~r = ~vi t +
1
~a t2 ;
2
∆~v = ~a t .
(4) Constant-Speed Circular Motion: for motion at constant speed v ≡ |~v | around a
circular trajectory of radius r.
The velocity vector ~v is always tangential to trajectory and perpendicular to acceleration
vector ~a: ~v ⊥ ~a.
The acceleration vector ~a always points towards the center of the circular trajectory.
Period T and frequency f of revolution, angular velocity ω, and orbital speed v:
T =
2πr
2π
1
=
=
f
v
ω
2π
v
=
T
r
ω = 2πf =
v = ωr = 2πf r =
2πr
T
Circular centripetal acceleration:
v2
= ω2r
r
Orbital angle ∆φ and arc of circumference ∆s covered during time interval ∆t:
a=
∆φ = ω ∆t =
v ∆t
∆s
=
r
r
∆s = v ∆t = ω r ∆t = ∆φ r
(5) Newton’s 2nd Law:
m~a = F~
(6) Kinetic Knergy (KE), Work, Work-KE-Theorem: K=kinetic energy of object of
mass m moving at speed v; W =work done by force F~ on an object moving/moved with
displacement ∆~r, with ∆~r pointing at an angle θ from F~ and 0o ≤ θ ≤ 180o ; ∆K = Kf −Ki =
change of kinetic energy due to work done by total force F~ :
1
K = m v2 ,
2
W = F ∆r cos θ ,
∆K = W .
(7) Energy Conservation Law for ∆K ≡ Kf − Ki and ∆U ≡ Uf − Ui :
Ki + Ui = Kf + Uf
or
∆K + ∆U = 0
(8) Mechanical Power: P =rate of work done by force F~ on an object moving at speed
~v , with ~v pointing at an angle θ from F~ and 0o ≤ θ ≤ 180o :
P = F v cos θ .
5
Physics 1112
Spring 2010
University of Georgia
Instructor: HBSchüttler
Algebra and Trigonometry
az 2 + bz + c = 0
sin θ =
opp
,
hyp
⇒
cos θ =
z=
adj
,
hyp
−b ±
√
b2 − 4ac
2a
tan θ =
opp
sin θ
=
adj
cos θ
sin2 θ + cos2 θ = 1
For very small angles θ (with |θ| 90o ):
sin θ ∼
= tan θ ∼
= θ (in radians)
Numerical Data
Acceleration of gravity (on Earth):
Speed of light in vacuum:
Biot-Savart’s constant:
Permittivity of vacuum:
Permeability of Vacuum:
Electron mass:
Proton mass:
c = 3.00 × 108 m/s
k = 8.99 × 109 Nm2 /C2
Coulomb’s constant:
Elementary charge:
g = 9.81m/s2
km ≡
µo
4π
= 1 × 10−7 Tm/A (exact)
o ≡ 1/(4πk) = 8.85 × 10−12 C2 /Nm2
µo ≡ 4πkm = 4π × 10−7 Tm/A (exact)
e = 1.60 × 10−19 C
me = 9.11 × 10−31 kg
mp = 1.67 × 10−27 kg
Other numerical inputs will be provided with each problem statement.
SI numerical prefixes:
y = yocto =10−24 , z = zepto =10−21 , a = atto =10−18 , f = femto =10−15 , p = pico =10−12 ,
n = nano =10−9 , µ= micro =10−6 , m = milli =10−3 , c = centi =10−2 , d = deci =10−1 ,
da = deca =10+1 , h = hecto =10+2 , k = kilo =10+3 , M = Mega =10+6 , G = Giga =10+9 ,
T = Tera =10+12 , P = Peta =10+15 , E = Exa =10+18 , Z = Zetta =10+21 , Y = Yotta =10+24 .
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