PHY
5**
Formula
Sheet
You MUST remember the formulas highlighted in this
formula sheet!!
2
3
4
5
÷(%&)
!"# % !"
&⎯⎯(
)*
×(%&)
!"# % !"
+⎯⎯, )*
Proof: by considering the units in the formula, - = 2 0 1 , we have
[ !"# % !" ] = 20 [)*] = 20 [ % !" ]
÷ #.%
!"/ℎ &⎯⎯( " ) &'
×*.,
45/ℎ +⎯⎯, 5 % !"
Proof: 1 45/ℎ =
1 5% !" =
(" - "...) /
(,. - ,. ) 0
!
2 3/
!"""
!
1
2 4
#" % #"
1
"
= 0.2785% !" = *., 5% !"
= 3.6 45/ ℎ ;
×
&
"5.
#?@!??% (°) &⎯⎯⎯( !"#C"D% (!"#)
×
"5.°
&
#?@!??%(°) +⎯⎯⎯, !"#C"D% (!"#)
Proof: 16 = 1 E
%&
*,.&
= 0.0175 !"# ; 1 !"# = 1 E
*,.&
%&
×"., ×".'!(
GHIJ?% (G) &⎯⎯⎯⎯⎯⎯⎯( ?K
÷"., ×".'!(
GHIJ?% (G) +⎯⎯⎯⎯⎯⎯⎯, ?K
Definition: 1 ?K = 1.6 × 10!"7 G
6
= 57.36 ;
+ powers
----
k 10*
M 10,
G 107
T 10"%
- powers
c 10!%
m 10!*
μ 10!,
n 10!7
p 10!"%
f 10!"8
+ power
----
Kilo
Mega
Giga
Tera
----
- power
centi
Milli
Micro
Nano
Pico
Femto
Control variable:
Dependent variable:
Independent variable:
Kept constant
Measured
Changed
Random Error:
Systematic Error:
Causing the measurements to differ from
the true value by a consistent amount each
time
Causing measurements to
spread about the true value
(usually by a different amount each time)
® Usually caused by the method of
observation & choice of instruments
® Cannot be corrected by repeats
® Cannot be corrected, but can be reduced by ® Can be confirmed using a different
repeats (& taking the mean)
technique/ method/ equipment/
instrument
® E.g. zero error (shifts all measurements
by the same amount)
7
Assuming acceleration (a) is constant, i.e. uniformly accelerated,
M = N + PQ
% = IR +
1
" R%
2
S % = I% + 2 " %
% =
"
%
(I + S)R
-----> unlikely that you’d need this in the exam
where s = displacement (+/—ive) (units: m) ;
where u, v = initial, final velocity (+/—ive) (units: m s—1 ) ;
where a = acceleration (+/—ive) (units: m s—1) ;
where t = time (≥ 0) (units: s) ;
8
U = VW + X
where y = y-coordinate (units: same as the units of the quantity plotted on the y-axis) ;
9:;<0 6= >
where m = gradient (units = 9:;<0 6= - ) ;
where x = x-coordinate (units: same as the units of the quantity plotted on the x-axis) ;
where c = y-intercept (units: same as units of y) ;
YZX [\]^Q_ = Z `
where arc length (units: m) ;
where r = radius (units: m) ;
where θ = (units: rad) ;
aNZbPX\ PZ\P = c d Z _ + c ( d Z? )
= XNZM\e fNZbPX\ PZ\P (XgZXNVb\Z\]X\ × _\g^_Q)
+ c × (Qhi/ jhQQhV fNZbPX\PZ\P)
where surface area (units: m2) ;
where r = radius of cylinder (units: m ) ;
where h = height of cylinder (units: m) ;
where volume of cylinder (units: m3) ;
9
aNZbPX\ PZ\P hb fi_\Z\ = k d Z?
lh[NV\ hb fi_\Z\ =
where surface area (units: m2) ;
where r = radius of sphere (units: m ) ;
where volume (units: m3) ;
fg]` ≈ `
Xhf` ≈ n
QP]` ≈ `
where θ = angle (units: rad) ;
Derivation:
10
@
A
d ZA
o = V X pq
energy transfer during heating & cooling
o = [ pV
energy transfer during change of state
where E = energy transferred for heating/change of state (units: J) ;
where m = mass of object heated/cooled in the energy transfer (units: kg ) ;
where c = specific heat capacity of the object heated/cooled (units: J kg—1 K—1) ;
where ΔT = change in temperature (units: K)
NB: you can use both degree Celsius oC and Kelvin (K) for the CHANGE in temperature,
because the temperature scales are only offset by 273, i.e where 0oC VS 0 K is defined, BUT,
you MUST use Kelvin (K) for temperatures!!!!!!);
where J = latent heat (of vaporisation/ condensation/solidification/fusion/sublimation)
(units: J kg—1) ;
where Δm = mass undergoing a change of state (i.e. solid<-->liquid<--->gas/ solid<-->gas)
(units: kg) ;
i l = ] r q = s tB q
il =
C
A
oD =
s V uuuu
X?
AEF
? G)
equation of state for an ideal gas
kinetic theory equation
molecular kinetic energy
where p = pressure (units: Pa) ;
where V = volume (units: m3 ) ;
where n = number of moles of gas (dimensionless) ;
where R = molar gas constant = 8.31 (units: J mol—1 K—1) ;
where T = temperature (units: K) ;
where N = number of gas molecules (dimensionless) ;
where 4H = Boltzmann constant = 1.38 × 10!%* (units: J K—1) ;
where m = rest mass of a gas molecule (units: kg) ;
where uuuu
v % = mean squared speed of the gas molecules (units: m2 s—2 ) ;
where wI = molecular KE (units: J) ;
where xJ = Avogadro constant = number of gas molecules in 1 mole of gas molecules =
6.02x1023 (units: mol—1) ;
11
y=V
∆L
∆M
=
∆N
∆M
i.e. force = rate of change of momentum
where F = force acting on a mass (units: N) ;
where m = mass of object acted on by the force (units: kg ) ;
where Δv = (units: m s—1) ;
where Δp = Δ(mv) (units: s) ;
where Δt = (units: s) ;
zhV\]Q = y e moment of a force
where moment/ torque about a point (units: N m) ;
where F = force (units: N) ;
where d = perpendicular distance between the point and the point of action of the force
(units: m) ;
12
oO = V ^ _
C
oD =
?
gravitational PE
V M?
{=yM
kinetic energy
mechanical power
where wP = gravitational PE (units: J) ;
where m = mass (units: m s—1 ) ;
where g = 9.81 (units: m s—2) ;
where h = height (units: m),
where wI = KE (units: J) ;
where v = speed (units: m s—1) ;
where P = mechanical power (units: W) ;
where F = applied force (units: N) ;
P=
L*
Q
= Z |?
centripetal acceleration
| =
M
Z
|=cdb
y =
R L*
Q
= V Z |?
where a = (units: m s—2) ;
where v = linear speed (units: m s—1 )
NB: usually refers to the speed along the circumference;
Where r = radius of the path taken in the circular motion (units: m) ;
Where ω = angular velocity (units: rad s—1)
i.e. the angular equivalent of the linear speed v ;
where f = frequency (units: Hz) ;
where F = centripetal force (units: N) ;
13
y =
S R+ R*
Q*
Newton’s 1st law of gravitation
where F = magnitude of the gravitational (attractive) force (units: N)
NB: in vector form, F ≤ 0 where a negative sign is added to the RHS to represent an
attractive force ;
where G = universal gravitational constant = 6.67 × 10!"" (units: N m2 kg—2 ) ;
where 5" , 5% = mass of interacting masses (via gravity) (units: kg) ;
where r = separation between the centre of mass CoM of the interacting masses (units: m) ;
e.g.
14
pU =
TU
fringe separation in double-slit interference
V
e fg]Ѳ = ] •
diffraction grating equation
where Δy = separation between fringes (formed at the screen) (units: m) ;
where λ = wavelength of light passing through the slits/grating (units: m) ;
where D = separation between the double-slit & the screen (units: m) ;
where a = separation between the slits (units: m) ;
where d = grating spacing = separation between each grating (units: m) ;
where Ѳ = angle that the ray makes with the perpendicular between the grating & the screen
(units: degrees o or rad) ;
where n = order of fringes = 0, 1, 2, 3, (dimensionless) ;
C
W
+
C
L
=
C
X
equation for a single lens
where u = object distance (units: m) ;
where v = image distance (units: m) ;
where f = focal length of the lens (units: m),
i.e. it’s a property of the lens, it’s constant unless we change the lens itself ;
15
y =
C
[+ [*
@ Y Ɛ,
Q*
Coulomb’s law
where F = electrostatic force between 2 point charges (units: N) ;
where Ɛ. = permittivity of free space = 8.85 × 10!"% (units: C2 N—1 m—2) ;
where Å" , Å% = electric charge of the point charges (units: C) ;
where r = separation between the point charges (units: m) ;
o=
o=
C
[
@ Y Ɛ,
Q*
\
]
electric fields strength due to a point charge
electric field between parallel plates (numerically)
where E = electric field strength (units: N C—1 or V m—1 ) ;
where Q = charge carried by the point charge that is causing the electric field (units: C) ;
where V = potential difference across the parallel plates (units: V) ;
where d = separation between the parallel plates (units: m) ;
16
r =
^_
resistance &resistivity
`
where R = resistance when electric current flows through a material (units: Ω) ;
where Ç = resistivity of a material (units: Ωm ), which is a property of the material ;
where J = length of the material / length of the path taken by the electric current through the
material (units: m) ;
where A = cross sectional area presented to the electric current as it flows through the
material (units: m2) ;
r = rC + r? (+ rA + r@ + … . . )
C
E
=
C
E+
+
C
E*
Ñ+
C
E-
+
C
E.
+ ⋯Ü
resistance in series
resistance in parallel
where R = equivalent resistance of a 2 resistors connected in series (units: Ω) ;
where á" , á% , ….. = resistance of individual resistors connected (units: Ω) ;
17
{ = à l = à? r
power in a circuit
l = à r ----> which is how we got from à l to à? r in the formula above
where P = power (units: W) ;
where â = current (units: A ) ;
where V = potential difference (units: V),
NB: the correct terminology is ‘potential difference’, NOT ‘voltage’!! Even though some of
your textbooks or teachers use the word ‘voltage’. We know they mean the same thing, and
some exams may accept it as a correct answer, ‘voltage’ is NOT the correct terminology!!! ;
where R = resistance (units: Ω) ;
y = ä ã M fg]Ѳ
where F = magnetic force acting on each moving charge (units: N) ;
where B = magnetic field strength that the charge is moving in (units: T ) ;
where Q = charge carried by the moving charge (units: C) ;
where v = drift velocity of the moving charge through the B-field (units: m s—1) ;
where θ = angle between the drift velocity of the moving charge and the direction of the Bfield (units: o or rad) ;
18
y = ä à å fg]Ѳ
where F = magnetic force acting on the current-carrying wire (units: N) ;
where B = magnetic field strength that the current-carrying wire is in (units: T ) ;
where â = current in the wire (units: A) ;
where L = length of the current-carrying wire INSIDE the B-field (units: m) ;
where θ = angle between the direction of the current and the direction of the B-field (units: o
or rad) ;
ä =
ça à
cdZ
where B = magnetic field at a perpendicular distance r from an infinitely long straight
current-carrying wire with current â (units: T) ;
where é. = permeability of free space = 40 × 10!b (units: H m—1) ;
where â = current in the wire, which is generating the B-field (units: A) ;
where r = perpendicular distance/ radial distance away from the straight wire (units: m) ;
19
ä =
c, G d
_
= ça ] à
where B = magnetic field at the CENTRE of a long solenoid i.e. along its axis (units: T) ;
where é. = permeability of free space = 40 × 10!b (units: H m—1) ;
where N = total number of turns of coil in the solenoid (dimensionless) ;
where â = current carried by the coils in the solenoid, which is generating the B-field (units:
A) ;
where J = total length of the solenoid (units: m) ;
where n = number of turns of coils per unit length in the solenoid (dimensionless) ;
Ɛ =s
pê
pQ
ê = äY
where ε = induced e.m.f in a conductor = rate of change of magnetic flux/ flux cutting
(units: V) ;
where N = number of turns in a coil (dimensionless) ;
where Δφ = change in magnetic flux (units: Wb) ;
where Δt = time interval (units: s) ;
where φ = magnetic flux (units: Wb) ;
where B = magnetic flux density = magnetic field (units: T or Wb m—2 ) ;
where A = area (units: m2) ;
20
\/
\0
≈
G/
G0
i.e. ratio of secondary voltage to primary voltage in a transformer
where Ke = potential difference across the Secondary coil (units: V) ;
where KP = potential difference across the Primary coil (units: V) ;
where xe = number of turns/ windings in the Secondary coil (dimensionless) ;
where xP = number of turns/ windings in the Primary coil (dimensionless) ;
21
s = sa \! f M
QC/? =
Y = ts
hi ?
f
law of radioactive decay
half-life & decay constant
activity & the number of undecayed nuclei
where N = number of undecayed radioactive nuclei remaining after a time t
(dimensionless) ;
where x. = number of undecayed radioactive nuclei at the beginning (i.e. at t = 0)
(dimensionless) ;
where k = decay constant = probability of a decaying occurring per unit time (units: s—1) ;
where t = time (units: s) ;
where R"/% = half-life = time taken for half of the radioactive nuclei to decay (units: s) ;
where A = activity of a radioactive source (units: Bq) ;
po = p( V X? )
mass-energy relationship/ energy-mass equivalence
where E = energy (units: J or eV) ;
where m = mass (units: kg or eV/c2 ) ;
where c = speed of light = 3 × 105 (units: m s—1) ;
22
Wave speed X = b •
where c = wave speed (units: m s—1) ;
where f = frequency (units: Hz ) ;
where λ = wavelength (units: m) ;
à =
j[
l =
k
Ɛ =
jM
[
l
[
Ɛ = à ( r + Z)
where â = current (units: A) ;
where Q = charge (units: C ) ;
where t = time (units: s) ;
where V = potential difference (units: V) ;
where W = work done (units: J) ;
where Ɛ = e.m.f. (units: V) ;
where E = energy (units: J) ;
where R = resistance (units: Ω) ;
where r = internal resistance (units: Ω) ;
23
] =
m
refractive index
m/
]C fg] `C = ]? fg] `?
fg]`n =
o*
o+
law of refraction
(for optically denser ]C > ]? optically less dense medium)
where c = wave speed (units: m s—1) ;
where n = refractive index (dimensionless) ;
where ve = speed in the medium (units: m s—1) ;
where D" , D% = refractive index (dimensionless) ;
where í" , í% = angle of incidence (units: o or rad) ;
where íp = critical angle (units:
q
or rad) ;
For a quadratic equation of the form P W? + j W + X = ì,
W =
− j ± √ j? − k P X
cP
NB: this is generally used to solve for the time t in the Equation of Motion % = I R + ½ " R %
\bbgXg\]XU =
24
WrsXW_ NqtsQ qWMNWM
NqtsQ uoNWM
ò =—
SvR
gravitational PE
Q
where U = gravitational PE (units: J) ;
where G = universal gravitational constant = 6.67 × 10!"" (units: N m2 kg—2 ) ;
where M, m = mass of interacting masses (via gravity) (units: kg) ;
where r = separation between the centre of mass CoM of the interacting masses (units: m) ;
{ = ö Y q@
Stefan’s law
where P = power output i.e. luminosity of a star (units: W) ;
where σ = Stefan constant = 5.67 x 10—8 (units: W m—2 K—4 ) ;
where A = surface area of the star (units: m2) ;
where T = absolute temperature of the star (units: K) ;
õ
jX
X,
õ≈
L
m
≈õ
jT
T,
õ
Doppler effect
where Δf = Doppler shift in frequency (units: Hz) ;
where 1. = frequency in the rest frame of the moving source (units: Hz ) ;
where v = velocity of the moving source (units: m s—1) ;
where c = speed of light (units: m s—1) ;
where Δλ = Doppler shift in wavelength (units: m) ;
where ú. = wavelength in the rest frame of the moving source (units: m) ;
25
C
?
oo = —
C
o*
ù
Vs MRVw ? = _ b — ê
"s #s @
?
$ % &a
?
û =—
•=
CA.x
o*
y
N
Einstein’s photoelectric equation
\l
=
y
RL
energy level equation for Hydrogen atom
de Broglie formula
i_hQh] \]\Z^U o = _ b =
_X
•
where me = mass of a photoelectron emitted = electron rest mass = 9.11 × 10!*" (units:
kg) ;
where S/z- = max. velocity that a photoelectron can be emitted with (units: m s—1 ) ;
where h = Planck constant = 6.63 × 10!*{ (units: J s) ;
where f = frequency of radiation incident on the metal surface (units: Hz) ;
where φ = work function (units: J) ;
where w: = energy of the nth energy level of a Hydrogen atom (units: eV) ;
where n = energy level = 1, 2, 3….. (dimensionless) ;
where †| = charge of an electron = 1.6 × 10!"7 (units: C) ;
where Ɛ. = permittivity of free space = 8.85 × 10!"% (units: C2 N—1 m—2 ) ;
where 1 eV = n. ° × nì!C} J of energy ;
` ≈
C.?? T
]
Rayleigh criterion (resolving power)
where Ѳ = min. angular separation of 2 objects such that they’re just resolvable (units:
rad) ;
where λ = wavelength (units: m) ;
where d = diameter of the aperture (units: m) ;
26
o=
~
`
illuminance
where E = illuminance (units: lx) ;
where φ = luminous flux falling on surface (units: lm ) ;
where A = surface area (units: m2) ;
[
M
= ¢
` (F1 — F2 )
]
ò=
where
Å
<
Ä
]
rate of energy transfer by conduction
thermal transmittance U-value
= rate of conduction during a heat transfer between the opposite ends of a body
(units: W) ;
where £ = thermal conductivity (units: W m—1 K—1 ) ;
where §Ç , §p = temperatures of the hotter (H) and colder (C) ends (units: T) ;
where A = cross section area of the body (units: m2) ;
where d = distance between the opposite end of the body (units: m) ;
where U = thermal transmittance = U-value (units: W m—2 K—1 ) ;
27
`≈
C.?? T
Rayleigh criterion (resolving power)
]
ih•\Z =
C
power of a lens
X
where Ѳ = min. angular separation of 2 objects such that they’re just resolvable (units:
rad) ;
where λ = wavelength (units: m) ;
where d = diameter of the aperture (units: m) ;
where power of a lens (units: D) ;
where f = focal length of the lens (units: m) ;
å = nì [h^
d
intensity level (dB)
d,
where L = relative intensity level (units: dB) ;
where log = logarithm to the base 10 ;
where â = intensity level (units: W m—2)
where â. = threshold intensity level of hearing (units: W m—2 ) ;
¶ = ßX
acoustic impedance
where Z = acoustic impedance (units: kg m—2 s—1) ;
where Ç = mass density of the medium that acoustic waves are travelling through (units: kg
m—3 ) ;
where c = speed of sound in the medium (units: m s—1) ;
28
à = àa \! c w transmitted intensity through a medium
where â = intensity of radiation transmitted through a medium after a distance of x (units:
W m—2) ;
where â. = intensity of radiation initially incident on the medium (units: W m—2 ) ;
where é = linear attenuation coefficient of medium (units: m—1), so the expression in the
exponential is dimensionless ;
where x = depth/ distance that the radiation has travelled in the medium (units: m) ;
29