Physics 4D Formula Sheet
Formulas needed for homework and quiz problems, compiled by David Eldon. For detailed explanations, refer to the longer study guide.
Physics 4D, Winter 2009, Professor Jenkins
Optics
n1 Sin@Θ1 D = n2 Sin@Θ2 D
c
Λ
cn =
; Λn =
n
n
IΘ = I0
EΘ 2
E0 2
Mirrors and lenses:
1
1
+
1
=
di
do
f
di
m=-
do
f > 0 for converging lens, concave (bowl) mirrior.
f < 0 for diverging lens, convex (ball) mirrior.
do > 0 for object on same side of mirror/lens as incoming light.
di > 0 for image on opposite side of lens as incoming light or in front of mirror
di > 0 means the image is real. di < 0 means that the image is virtual.
1
f
= Hn - 1L
1
Hfor focal length of spherical lens, convex surfaces have positive RL
1
+
R1
R2
Hfor focal lengh of spherical mirrorL
r
f=
2
Constructive interference from reflections off of two parallel surfaces (like the soap bubble):
1
m+
Λn = 2 t
2
Phase shift between reflections from parallel surfaces:
t
∆=Π+4Π
Λn
Phasors:
Draw the amplitude of one electric field as a vector in phase space, then draw the amplitude of the next vector starting from the tip of the first one.
The angle between them should be the phase shift between them, ∆. The length of a vector from the tail of first first field vector to the tip of the last
vector will be the amplitude of the resulting wave.
Double slit interference:
IΘ = I0 CosB
∆
2
F
2
, where ∆ =
HMaximaL
d Sin@ΘD = m Λ
1
d Sin@ΘD = m +
Λ
2
HMinimaL
2Π
d Sin@ΘD
Λ
2
Single slit diffraction:
Sin@Β  2D
2
2
IΘ = I0
Β=
, where
2Π
D Sin@ΘD
Λ
HMaxima HΘ = 0 is the central maximumLL
1
D Sin@ΘD = m +
Λ
2
HMinima, m ¹ 0 !L
D Sin@ΘD = m Λ
Single and double slit effects together:
Sin@Β  2D
2
2
IΘ = I0
∆
CosB
2
F
2
Diffraction grating:
mΛ
Sin@ΘD =
d
Λ
R=
HResolving powerL
=Nm
DΛ
Circular hole diffraction:
Θ=
1.22 Λ
D
Polaroids (aka polarizing filters) (Divide by 2 if light coming into the first filter is unpolarized.):
I = I0 HCos@ΘDL2
Tidal Accelerations
Things get stretched in the direction away from the source of gravity and get compressed in the direction around the source of gravity. To derive the
Ó
`
equation for this, just remember the equation for gravitational acceleration IFg = GrM2 m rM and subtract the two acceleration vectors.
Ó
Ó
Ó
arelative = a1 - a2 = G M
`
r1
r1 2
-
`
r2
r2 2
Relativity
HDsL2 = HDxL2 - Hc DtL2
HDs 'L2 = HDsL2
Hc DΤL2 = Hc DtL2 - HDxL2
Lorentz Transformations (works on general 4-vector if ct,x,y,z replaced with 0,1,2,3 components of 4-vector):
v
ct = Γ
x ' + ct '
c
v
x = Γ x' +
ct '
c
y = y'
z = z'
1
Γ=
1 - Hv  cL2
Addition of velocities (you could replace x with parallel and y with perpendicular for more generality):
ux =
ux ' + v
1+
ux ' v
c2
uy '
uy =
Γ J1 +
ux ' v
c2
N
Length contraction and time dialation:
Proper length is the longest length. Proper time is the shortest time.
3
L
L' =
Γ
Dt ' = Γ Dt
Accelerating particle (velocity as function of time) :
Τ=á
1-
Hv@tDL2
ât
c2
Momentum and Energy :
p=Γmv
E=
m2 c4 + É p È2 c2 =
m2 c4 + px 2 c2 + p y 2 c2 + pz 2 c2 = Γ m c2
E2 = m2 c4 + p2 c2
KE =
m2 c4 + p2 c2 - m c2 = HΓ - 1L m c2
4 - Vectors :
rΜ = H ct, x, y, z L = I ct, Ó
rM
Ó
E
E
Μ
p = I €c , px , py , pz M = I €c , p M = H Γ m c, px , py , pz L = H Γ m c, Γ m vx , Γ m vy , Γ m vz L
pΜ × pΜ =
E
2
- p2
c
Light pressure (f is the fraction of light reflected back TOWARDS the source, F is the incoming power flux):
F = H1 + fL
Power
c
Pressure = H1 + fL F
1
c