Copyright © 2007 The Open University WEB 96146 9

S207

STANDARD EQUATIONS AND

CONSTANTS

This complete list of constants, mathematical formulae and physics

equations is included for reference. It may be useful as an aid to your

memory but please bear in mind that many of the entries will not be

needed in this examination.

2

Useful constants

magnitude of the acceleration

due to gravity on Earth g = 9.81 m s−2

Newton’s universal

gravitational constant G = 6.673 × 10−11 N m2 kg−2

Avogadro’s constant Nm = 6.022 × 1023 mol−1

Boltzmann’s constant k = 1.381 × 10−23 J K−1

molar gas constant R = 8.314 J K−1 mol−1

permittivity of free space ε0 = 8.854 × 10−12 C2 N−1 m−2

1/4πε0 = 8.988 × 109 N m2 C−2

permeability of free space µ0 = 4π × 10−7 T m A−1

speed of light in vacuum c = 2.998 × 108 m s−1

Planck’s constant h = 6.626 × 10−34 J s

" = h/2π = 1.055 × 10−34 J s

Rydberg constant R = 1.097 × 107 m−1

Bohr radius a0 = 5.292 × 10−11 m

atomic mass unit amu (or u) = 1.6605 × 10−27 kg

charge of proton e = 1.602 × 10−19 C

charge of electron −e = −1.602 × 10−19 C

electron rest mass me = 9.109 × 10−31 kg

charge to mass ratio of

the electron −e/me = −1.759 × 1011 C kg−1

proton rest mass mp = 1.673 × 10−27 kg

neutron rest mass mn = 1.675 × 10−27 kg

radius of the Earth 6.378 × 106 m

mass of the Earth 5.977 × 1024 kg

mass of the Moon 7.35 × 1022 kg

mass of the Sun 1.99 × 1030 kg

average radius of Earth orbit 1.50 × 1011 m

average radius of Moon orbit 3.84 × 108 m

3

Mathematical formulae

2 4

2

b b acx

a

− ± −=

2π rad = 360°

sin( 2 ) sin( )θ θ+ π =

cos( 2 ) cos( )θ θ+ π =

sin( ) sin( )θ θ− = −

cos( ) cos( )θ θ− =

2 2sin ( ) cos ( ) 1θ θ+ =

sin( )tan( )

cos( )

θθ

θ=

a ·b = ab cos θ

a ·b = axbx + ayby + azbz

a × b = (aybz − azby, azbx − axbz, axby − aybx)

|a × b | = ab sin θ

If d

dt

α=

v

v , then v(t) = v0 eα t

ex ey = ex+y

(ex)y = exy

e−x = 1/ex

logeex = x

1

N

i i

i

x p x

=

⟨ ⟩ =∑

sphere surface area = 4πr2

sphere volume = 34

3

rπ

Derivatives

[A, n, k and ω are constants; x, y and z are

functions of t]

xd

d

x

t

A 0

tn ntn−1

sin(ω t) ω cos(ω t)

cos(ω t) −ω sin(ω t)

ekt kekt

Ayd

d

yA

t

y + zd d

d d

y z

t t+

4

Book 2 Describing motion

x xs t= v ( constant)

x=v

1 2

2x x xs u t a t= +

x x xu a t= +v

2 2 2x x x x

u a s= +v

1

2( )

x x xs u t= +v

t= +u av

1 2

2t t= +s u a

arcs R θ= Δ

d 2 rad

dt T

θω

π= =

2 rr

Tω

π

= =v

2

2a rr

ω ω= = =

v

v

22 f

Tω

π

= π =

( ) sin( )x t A tω φ= +

2

2

2

d ( )( )

d

x tx t

t

ω= −

2 2

2 21

x y

a b+ =

2 21

e a ba

= −

2

3

TK

a=

5

Book 3 Predicting motion

F = ma

W = mg

1 2

212

ˆ

Gm m

r

−

=F r

Fmax = µstaticN

F = µslideN

F = 6πηRv

2

2m

F mrr

ω= =

v

Fx = −ksx

s

22

mT

kω

π

= = π

22

lT

gω

π

= = π

1 2trans 2E m= v

W = F · s

Egrav = mgh

1 2str s2

E k x=

1 2grav

Gm mE

r

−

=

potd

dx

EF

x

−

=

P = d

d

W

t = F · v

x(t) = (A0 e−t /τ) sin(ω t + φ)

2 total stored energy

average energy loss per oscillationQ

π ×

=

0

2 2 20

/

( ) ( / )

F mA

b mω Ω Ω2

=

− +

p = mv

d

dt

=

pF

Γ = r × F

v = ω × r

d

dt

=

ω

α

2i i

i

I m r=∑

1 2rot 2

E Iω=

1 12 2kin CMCM2 2

E M I ω= +v

W = ΓΔθ

P = Γ ·ω

l = r × p

L = Iω

Γ =d

dt

L

6

Book 4 Classical physics of matter

(number density)M

mV

ρ = = ×

FP

A

⊥=

3m r m10 kgM M N m−

= × =

1 d

d

V

V Tα =

1 d

d

V

V Pβ = −

PV nRT NkT= =

2 2

f fU nRT NkT= =

3trans 2

E kT⟨ ⟩ =

p = A e−E /kT

2 2( ) exp( /2 )f B m kT= −v v v

3 / 2

42

mB

kT

= π π

mp

2kT

m=v

8kT

m⟨ ⟩ =

πv

rms

3kT

m=v

/( ) e E kTg E C E −

=

3 / 22 1

CkT

=

π

1mp 2

E kT=

32

E kT⟨ ⟩ =

tot

2

fE kT⟨ ⟩ =

W = −PΔV

ΔU = Q + W

QC

T=Δ

2V

fC nR=

P VC C nR= +

21

P

V

C

C fγ = = +

PV = I

PV Aγ=

revQ

ST

Δ =

2 2

2 1 e e

1 1

log logV P

P VS S C C

P V

− = +

S = k logeW

c

h h

1W T

Q Tη = = −

c c

h c

Q T

W T Tκ = =

−

AP P g zρ= +

( ) (0)exp( / )P z P z λ= −

kT

mgλ =

2 1

1 2

A

A=

v

v

1 2

2constantP ghρ ρ+ + =v

F Ax

ηΔ

=Δ

v

0 0LRe

ρ

η=

v

7

Book 5 Static fields and potentials

1 2grav 2

ˆ

Gm m

r= −F r

1 2

el 20

1ˆ

4

q q

rε

=

π

F r

Fgrav (on m at r) = mg(r)

Fel (on q at r) = qe(r)

20

ˆ( )4

Q

rε

=

π r re

1 2grav

Gm mE

r= −

1 2

el

04

q qE

rε

=

π

Eel (with q at r) = qV(r)

0

1( )

4

QV

rε

=

π r

d ( )

dx

V

x= −

r

E

qC

V=

C = εr ε0A

d

2 2

el2 2 2

qV CV qE

C= = =

d

d

qi

t=

i neA= v

R| |

| |

VR

i=

LR

A

ρ=

V iR=

eff i

i

R R=∑

∑=i iRR

11

eff

2

2V

P iV i RR

= = =

EMFV V ir= −

/0 e

tq q τ−

=

/0 e

ti i τ−

=

τ = RC

Fm = q [v × B(r)]

0( )2

iB r

r

µ=

π

0

centre2

NiB

R

µ=

NiB

l

µ0=

F = q[e(r) + v × B(r)]

C

1

2

q Bf

m

| |=

π

perpC

mR

q B

| |=

| |

v

H

iV B

nqt=

Fm = l[i × B(r)]

mF Bil=

0 1 2

2

F i i

l d

µ=

π

iABΓ =

8

Book 6 Dynamic fields and waves

cosABφ θ=

ind

d ( )( )

d

tV t

t

φ| | =

ind

d ( )( )

d

i tV t L

t| | =

2ANL

l

µ=

1 2mag 2

E Li=

bat

d ( )( )

d

i tV L i t R

t− =

( )bat /( ) 1 e Rt LV

i tR

−

= −

Vmax

= imaxωL

max

max

iV

Cω=

1

LCω =

1

2

d ( )( )

d

i tV t M

t| | =

1 2N N AM

l

µ=

2

2 1

1

( ) ( )N

V t V tN

| | = × | |

sin( )y A kx tω= −

2k

λ

π

=

2

Tω

π

=

fk

ωλ= =v

TF

µ=v

obs emf fV

= ±

v

v

obs em

Vf f

± =

v

v

( , ) 2 cos( )sin( )y x t A t kxω=

2L n

λ =

c f λ=

1 2

2 1

sin

sin

i n

r n= =

v

v

1 2

crit

2 1

sinn

in

= =

v

v

sinn

n dλ θ=

sinw

λθ =

1 1 1

u f+ =

v

m

u

=

v

P = 1/f

Ptotal = P1 + P2 + P3 + …

IM

OB

Mα

α

=

o

e

fM

f=

light-gathering power =

2

o

p

D

D

maximum light-gathering power

=

22

o o

e e

D f

D f

=

exposure = IΔt

-number f

FD

=

(Continued overleaf)

9

0

2 21 /

TT

V c

ΔΔ =

−

02

21

LL

V

c

=

−

( )x x Vtγ′ = −

2( / )t t Vx cγ′ = −

2

2

1

1V

c

γ =

−

21

x

x

x

V

V

c

−′ =

−

v

v

v

2 21

m

c

=

−

p

v

v

2 2

d d

d d 1

m

t t c

= = −

pF

v

v

2

tot2 21

mcE

c

=

−v

Emass = mc2

2

2trans

2 21

mcE mc

c

= −

−v

2 2 2 2 4totE p c m c= +

E = cp

10

Book 7 Quantum physics: an introduction

E = hf

1 2e max2

m hf φ= −v

mevr = n"

1

2n

EE

n

| |= −

dB

h

pλ =

2x

x pΔ Δ ≥"

2E tΔ Δ ≥

"

( )2

tot pot2 2

d 2( ) 0

d

mE E x

x

ψψ+ − =

"

2 2

kin2

kE

m=

"

p = "k

2 2

tot28

n hE

mD=

2

2 2 2tot 1 2 32

( )8

hE n n n

mD= + +

P = |ψ (x1) |2Δx

P = |ψ1(r) |2ΔV = |ψ1(r) |2 4πr2Δr

4e

tot2 2 2 2

0

1 13.6eV

8

m eE

n h nε

= − = −

( 1)L l l= + "

Lz = ml"

( 1)S s s= + "

Sz = ms"

11

Book 8 Quantum physics of matter

3/ 2

/2 1( ) e E kTN

G E EkT

−

= ×

π

( )D E B E=

3

3 2

3

2(2 )

LB m

h

π

=

B ( )

1( )

e 1E kTF E

µ−=

−

B

1( )

e 1E kTF E =

−

FF ( )

1( )

e 1E E kTF E

−

=

+

Dp(E) = CE2

2p /

1( )

e 1E kTG E CE= ×

−

C = 8πV/h3c3

N = 2.4C(kT)3

4

4( )15

U C kTπ

=

3

UP

V=

m

m

zNn

V=

e( )D E B E′=

3 2e

3

4(2 )

VB m

h

π′ =

Fe ( ) /

1( )

e 1E E kTG E B E

−

′= ×+

2 32

F

e

3

8

h nE

m

= π

3F5

U NE=

2F5

P nE=

d

d

NN

tλ= −

N(t) = N0 exp(−λt)

[END OF BOOKLET]