Equations sheet for FYS1120

September 26, 2013

Electric elds

Coulomb's law

F =1

40

q1q2r2

r^

E =1

40

q

r2r^

E =1

40

ZV

r2r^ d

Dipoles

= B = pE

= IA p = qd

UB = B UE = p E

Potential, energy and work

Wa!b = Ua Ub

U =q040

Xi

qiri

V =U

q0=

1

40

Xi

qiri

V =1

40

Zdq

r

VB VA = Z BA

E dl

rV = E

Energy density in electromagnetic eld

u =1

2

0E

2 +1

0B2

Energy stored in solenoid and capacitor:

UB =1

2LI2; UE =

1

2

Q2

C

Maxwell's equations

In general

r E = 0

IS

E dA = Qencl0

r B = 0IS

B dA = 0

rE = @B@t

IL

E dl = dBdt

rB = 0J+ 00 @E@tI

L

B dl = 0 Ic + 0

dEdt

!encl

In matter

r D = fIS

D dA = Qfenc

r B = 0IS

B dA = 0

rE = @B@t

IL

E dl = ddt

ZS

B dA

rH = Jf + @D@t

IL

H dl = Ifenc +d

dt

ZS

D dA

Denitions

D = 0E+P

H =1

0BM

In linear media

P = 0eE D = E

M = mH H =1

B

Lorentz force

F = q v B+ q E

1

Magnetism

Flux:

=

ZB dA

Magnetic force on a conductor:

F = IlB

Faraday's law and emf

E = dBdtI

E dl = dBdt

All or parts of a closed loop moves in a B eld:

E =I(v B) dl

Biot-Savart law

B(r) =04

qv rr2

B(r) =04

I

Zdl r^r2

Self inductance & Mutual inductance

L =NBi

; E = Ldidt

M =N2B2i1

=N1B1i2

E2 = M di1dt

E1 = M di2dt

Continuity of magnetic uxIS

0H dA = 0 :

over a closed surface.

Capacitor

C =Q

V= 0r

A

d

Capacitors in series:

1=Ceq = 1=C1 + 1=C2 + :::

Capacitors in parallel:

Ceq = C1 + C2 + :::

Resistor

R =V

I=L

A

Resistors in series:

Req = R1 +R2 + :::

Resistors in parallel:

1=Req = 1=R1 + 1=R2 + :::

Circuits

Current:

I =dQ

dt= njqjvdA

Eect:

P = V I :

Over ohmic resistance:

P = RI2 =V 2

R:

RC circuit

Charging capacitor in RC-circuit:

q = ECh1 e(t=(RC))

iDecharging capacitor:

q = Q0e(t=(RC))

RL circuit

E LdIdtRI = 0

I =ER

h1 et(R=L)

iWithout emf:

LdIdtRI = 0

I = Imet(R=L)

LC circuit

q

C= L

dI

dt(1)

d2q

dt2 qLC

= 0 (2)

! q = Qm cos(!0 + ) (3)

2

RCL

d2q

dt2+R

L

dq

dt+

1

LCq = 0 (4)

q = Qmet= cos(!t+ )

Qm and are dependent on the initial conditions.

! =

s1

LCR

2L

2=

r!20

1

2

Driven RCL

Complex current, voltage, impedance with inductive andcapacitive reactance.

I^ =VmZei!t V^ = Vmei!t

Irms =Imp2

Vrms =Vmp2

We have

Ld2I^

dt2+R

dI^

dt+

I^

C= i!Vme

i!t

Phase dierence

tan =XL XC

R

Power of RCL

P = RI2rms = VrmsIrms cos

Impedance

Z =pR2 + (XL XC)2 =

rR2 + (!L 1

!C)2

X = XL XC

Z = R+ jX =G jBG2 +B2

Z =VmIm

Reactance

Capacitive

XC = 1!C

; B = !C

Inductive

XL = !L; B = 1!L

Admittance

Y =1

Z= G+ jB =

R jXR2 +X2

TransformersV 2

V1=N2N1

V1I1 = V2I2

Units

Henry:

H =m2 kgs2 A2 =

J

A2=

Wb

A=

V sA

(5)

=J/C sC/s

=J s2C2

=m2 kgC2

= s (6)

Ampere:

A =C

s

Tesla:

T =V sm2

=N

A m =Wb

m2=

kg

C s =kg

A s2

1 Constants

Proton massmp = 1:67 1027 kg

Proton charge

qp = 1e = 1:602 1019C

Electron mass

me = 9:11 1031 kg

Electron charge

qe = 1e = 1:602 1019C

Electrical permittivity in vacuum

0 = 8:85 1012C2N1m2

Gravitational constant

G = 6:67 1011m3kg1s2

Magnetic permeability

0 = 4 107 Tm=A

Speed of light

c =1p00

3

Constants