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.

2Useful constants

magnitude of the acceleration

due to gravity on Earth g = 9.81 m s2

Newtons universal

gravitational constant G = 6.673 1011 N m2 kg2

Avogadros constant Nm = 6.022 1023 mol1

Boltzmanns constant k = 1.381 1023 J K1

molar gas constant R = 8.314 J K1 mol1

permittivity of free space 0 = 8.854 1012 C2 N1 m2

1/40 = 8.988 109 N m2 C2

permeability of free space 0 = 4 107 T m A1

speed of light in vacuum c = 2.998 108 m s1

Plancks constant h = 6.626 1034 J s

" = h/2 = 1.055 1034 J s

Rydberg constant R = 1.097 107 m1

Bohr radius a0 = 5.292 1011 m

atomic mass unit amu (or u) = 1.6605 1027 kg

charge of proton e = 1.602 1019 C

charge of electron e = 1.602 1019 C

electron rest mass me = 9.109 1031 kg

charge to mass ratio of

the electron e/me = 1.759 1011 C kg1

proton rest mass mp = 1.673 1027 kg

neutron rest mass mn = 1.675 1027 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

3Mathematical formulae

2 4

2

b b acx

a

=

2 rad = 360

sin( 2 ) sin( ) + =

cos( 2 ) cos( ) + =

sin( ) sin( ) =

cos( ) cos( ) =

2 2sin ( ) cos ( ) 1 + =

sin( )tan( )

cos( )

=

a b = ab cos

a b = axbx + ayby + azbz

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

|a b | = ab sin

If d

dt=

v

v , then v(t) = v0 e t

ex ey = ex+y

(ex)y = exy

ex = 1/ex

logeex = x

1

N

i i

i

x p x

=

=

sphere surface area = 4r2

sphere volume = 34

3

r

Derivatives

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

functions of t]

xd

d

x

t

A 0

tn ntn1

sin( t) cos( t)

cos( t) sin( t)

ekt kekt

Ayd

d

yA

t

y + zd d

d d

y z

t t+

4Book 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 =

d 2 rad

dt T

= =

2 rr

T

= =v

2

2a rr

= = =v

v

22 f

T

= =

( ) sin( )x t A t = +

2

2

2

d ( )( )

d

x tx t

t

=

2 2

2 21

x y

a b+ =

2 21

e a ba

=

2

3

TK

a=

5Book 3 Predicting motion

F = ma

W = mg

1 2

212

Gm m

r

=F r

Fmax = staticN

F = slideN

F = 6Rv

2

2m

F mrr

= =

v

Fx = ksx

s

22

mT

k

= =

22

lT

g

= =

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 et /) sin( t + )

2 total stored energy

average energy loss per oscillationQ

=

0

2 2 20

/

( ) ( / )

F mA

b m 2=

+

p = mv

d

dt=

pF

= r F

v = r

d

dt=

2i i

i

I m r=

1 2rot 2

E I=

1 12 2kin CMCM2 2

E M I = +v

W =

P =

l = r p

L = I

=d

dt

L

6Book 4 Classical physics of matter

(number density)M

mV

= =

FP

A

=

3m r m10 kgM M N m

= =

1 d

d

V

V T =

1 d

d

V

V P =

PV nRT NkT= =

2 2

f fU nRT NkT= =

3trans 2

E kT =

p = A eE /kT

2 2( ) exp( /2 )f B m kT= v v v

3 / 2

42

mB

kT

=

mp

2kT

m=v

8kT

m =

v

rms

3kT

m=v

/( ) e E kTg E C E =

3 / 22 1

CkT

=

1mp 2

E kT=

32

E kT =

tot

2

fE kT =

W = PV

U = Q + W

QC

T=

2V

fC nR=

P VC C nR= +

21

P

V

C

C f = = +

PV = I

PV A =

revQ

ST

=

2 2

2 1 e e

1 1

log logV PP V

S S C CP V

= +

S = k logeW

c

h h

1W T

Q T = =

c c

h c

Q T

W T T = =

AP P g z= +

( ) (0)exp( / )P z P z =

kT

mg =

2 1

1 2

A

A=

v

v

1 2

2constantP gh + + =v

F Ax

=

v

0 0LRe

=

v

7Book 5 Static fields and potentials

1 2grav 2

Gm m

r= F r

1 2

el 20

1

4

q q

r=

F r

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

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

20

( )4

Q

r

=

r re

1 2grav

Gm mE

r=

1 2

el

04

q qE

r=

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

0

1( )

4

QV

r

=

r

d ( )

dx

V

x=

r

E

qC

V=

C = r 0A

d

2 2

el2 2 2

qV CV qE

C= = =

d

d

qi

t=

i neA= v

R| |

| |

VR

i=

LR

A

=

V iR=

eff i

i

R R=

=i iRR

11

eff

2

2V

P iV i RR

= = =

EMFV V ir=

/0 e

tq q =

/0 e

ti i =

= RC

Fm = q [v B(r)]

0( )2

iB r

r

=

0

centre2

NiB

R

=

NiB

l

0=

F = q[e(r) + v B(r)]

C

1

2

q Bf

m

| |=

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

=

iAB =

8Book 6 Dynamic fields and waves

cosAB =

ind

d ( )( )

d

tV t

t

| | =

ind

d ( )( )

d

i tV t L

t| | =

2ANL

l

=

1 2mag 2

E Li=

bat

d ( )( )

d

i tV L i t R

t =

( )bat /( ) 1 e Rt LV

i tR

=

Vmax

= imaxL

max

max

iV

C=

1

LC =

1

2

d ( )( )

d

i tV t M

t| | =

1 2N N AM

l

=

2

2 1

1

( ) ( )N

V t V tN

| | = | |

sin( )y A kx t=

2k

=

2

T

=

fk

= =v

TF

=v

obs emf fV

=

v

v

obs em

Vf f

=

v

v

( , ) 2 cos( )sin( )y x t A t kx=

2L n

=

c f =

1 2

2 1

sin

sin

i n

r n= =

v

v

1 2

crit

2 1

sinn

in

= =

v

v

sinn

n d =

sinw

=

1 1 1

u f+ =

v

m

u

=

v

P = 1/f

Ptotal = P1 + P2 + P3 +

IM

OB

M

=

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 = It

-number f

FD

=

(Continued overleaf)

90

2 21 /

TT

V c

=

02

21

LL

V

c

=

( )x x Vt =

2( / )t t Vx c =

2

2

1

1V

c

=

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 = v

mevr = n"

1

2n

EE

n

| |=

dB

h

p =

2x

x p "

2E t

"

( )2

tot pot2 2

d 2( ) 0

d

mE E x

x

+ =

"

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 = | (x1) |2x

P = |1(r) |2V = |1(r) |2 4r2r

4e

tot2 2 2 2

0

1 13.6eV

8

m eE

n h n

= =

( 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

=

( )D E B E=

3

3 2

3

2(2 )

LB m

h

=

B ( )

1( )

e 1E kTF E

=

B

1( )

e 1E kTF E =

FF ( )

1( )

e 1E E kTF E

=

+

Dp(E) = CE2

2p /

1( )

e 1E kTG E CE=

C = 8V/h3c3

N = 2.4C(kT)3

4

4( )15

U C kT

=

3

UP

V=

m

m

zNn

V=

e( )D E B E=

3 2e

3

4(2 )

VB m

h

=

Fe ( ) /

1( )

e 1E E kTG E B E

= +

2 32

F

e

3

8

h nE

m

=

3F5

U NE=

2F5

P nE=

d

d

NN

t=

N(t) = N0 exp(t)

[END OF BOOKLET]