The Magnetic Field

The force on a charge q moving with a velocity F

v

)( BvEqF

The magnitude of the force

sinqvBF

sec)//(][ meterCoulombNewtonsB

gausssmCNewtonmwteslaT 42 10//1/1)(1

TGaussBEarth4101

left-hand ruleright-hand rule

Motion in magnetic field

1) Uniform , B

Bv

2) Uniform , B

Bv

3) Nonuniform B

0E

BvqF

qvBFBv

amF

0 FmaF rr

r

vmmrqvB

22 ri

qB

mvr

The angular velocity

m

qB

qBmvv

r

v

0|| FBv

The angular velocity (cyclotron frequency )

m

qB

qBmvv

r

v

2

f

does not depend on velocity!

The force is always perpendicular to velocity, so it cannot change the magnitude of the velocity, only its direction. The work done by the magnetic force is zero! Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed.

Electron motion in a microwave oven

A magnetron in a microwave oven emits electromagnetic waves with frequency f=2450 MHz. What magnetic field strength is required for electrons to move in circular paths with this frequency?

Using Crossed and Fields E

B

Velocity selector

0 qEqvB

vBE

B

Ev independent of the mass of the particle!

Mass spectrometer

qB

vmR 1

1

B

Ev

21

1 qB

EmR

22

2 qB

EmR

Thomson’s e/m experiment

1897: Cavendish Laboratoryin Cambridge, England

m

eVveVmv

2

2

1 2

B

Ev

2

2

2

2

VB

E

m

e

m

eV

B

E

Current carrying wires

1820 Hans Christian Oersted

Hans Christian Ørsted

Gauss’s Law

The total flux of electric field out of any closed surface is equal to the charge contained inside the surface divided by .0

S

enclosedQSdE

0

Conductors and insulators

Charges reside at the surface of the conductor

Conductor

E=0+

+ ++

+

+++

++

+++

+

+

Electric field of a ball of charge

30

20

4

1

4

1

R

rQERr

r

QERr

Q

Electric field outside of a charged sphere is exactly the same as the electric field produced by a point charge, located at the center of the sphere, with charge equal to the total charge on the sphere.

Insulating sphere with charge Q uniformly spread throughout the volume

A

E

r

204

1

r

QEAr

rA

QEAr

304

A

Conducting sphere with charge Q

A

E

r

204

1

r

QEAr

0 EAr

V

r

A

A

Q

04

1

r

QVAr

04

1

A

QVAr

04

1

A Charged, Thin Sheet of Insulating Material

++

+

+

++

+

++

++

02

E

A field in a cavity of a conductorFaraday’s cage

Electric field near a surface of a conductor

al

cap

EaEdSSdE

0a

Ea 0

E

A

QLL

A

ALbottomVtopV

000

)]()([

0AQL

V

The capacitance is:

L

A

AQLQ

V

QC 0

0

Spherical capacitor; Cylindrical capacitor

Capacitors in series: ...1111

321

CCCCtot

Capacitors in parallel: ...321 CCCCtot

22

2

1

2

1Q

CCVW

faradC ][

If the capacitors were initially uncharged,

Current Density

S

Sdji

Consider current flowing in a homogeneous wire with cross sectional area A.

jAdSjjdSSdjiA A A

A

ij for j =Const only!

Current, Ohm’s Law, Etc.

dt

dQi

)(;:' VoftindependenConstRi

VRLawsOhm

A

lR

Ej

jE

For steady state situation

1.Kirchhoff’s junction rule: The algebraic sum of the currents into any junction is zero.

2.Kirchhoff’s loop rule: The algebraic sum of the potential differences in any loop must be zero.

0 Sdj

0 rdE

Joule’s Law

R

VRiViP

22

Resistors in parallel:

Resistors in series:

21 RRR

21

111

RRR