Experimental Physics EP2a · When a charged particle moves parallel to the magnetic field lines,...

Experimental Physics II - Magnetic field 1

Experimental Physics EP2a

Thermodynamics & Electricity

– Magnetic field –Isolated charges, conductors

https://bloch.physgeo.uni-leipzig.de/amr/


Some history

Pierre Pelerin de Maricourt 1269

William Gilbert1600

Gian Domenico Romagnosi

1820, 11 September

1802

André-Marie Ampère

Hans Christian Ørsted 1819

1820s Michael Faraday Joseph Henry

1865 James Maxwell

A Dynamical Theory of the Electromagnetic Field


Experimental observations

The magnitude of the magnetic force on a moving charge is proportional to its speed |v| and to its charge q.

The magnitude and the direction of this force depends on the magnitude |B| and the direction of the magnetic field.

+

When a charged particle moves parallel to the magnetic field lines, the force F exerted upon the particle is zero.

However, if the angle between the magnetic field and the particle velocity is not zero, the force is perpendicular to both v and B.

If two charged particles with opposite charges move parallel to each other, the forces exerted upon the particles will be in opposite directions.

The magnitude of the magnetic forces will be proportional to sin(q), where q is the angle between the directions of v and B.

BvqFB!!!

´=


How to probe magnetic field

+

E!

ò= sdFWE!!

0=BW

Magnetic field can alter only the direction of a charged particle moving in this field, but not the speed or kinetic energy of the particle.

+

B!

dtsdF!!

^

BvqFB!!!

´=

[ ] TAmN

smC

Nº=

×=B

G10T1 4=

( ) ( ) ( )baccabcba!!!!!!!!!×-×=´´

2^

^´=qvvFB!!!


Conductors

B!

I

-v!

I

F!

BvQF dB

!!!´=

( )BvnALqF dB

!!!´=

tQID

= dvLQ

= dnAqv=

( )BLIFB!!!

´=

tvdD

L

Aq

qq

B!

dv!


Arbitrarily shaped wires

sd!

( )BLIFB!!!

´=BsIdFd B

!!!´=

ò ´=b

aB BsdIF

!!! )(sBB !!!=

0BB!!

=

0BsdIFb

aB

!!!´÷÷ø

öççè

æ= ò 0BLI

!!´=

L!

042 == FF!!

01 ILBF =!

3F!

-=å =loop

iF 0!

1

2

3

4

I0B!

L

1L

2

3

I

0B!

q R

1

2


Maximal torque on a loop

0IaBF =

abI

F

F

O

( ) Frdtprd !!!!

´=´

=dtLd!

!=t

bFbFnet 21

21 +=t 0IabB= 0IAB=

( )BLIFB!!!

´= ( ) ( ) ( )baccabcba!!!!!!!!!×-×=´´

( ) ( )[ ]LrBBrLI!!!!!!!×-×=t

Ir!Ld!

( ) ( )[ ]LdrBBrLdId!!!!!!!

×-×=t

( ) ( )[ ]LdrBBrLdId!!!!!!!

×-×=t

ò ^^ = dLIBRd at cosI

^t

ò=p

aa2

0

22 cos dIBR 2RIBp=


BAI!!!

´=t

n̂

n̂AA =!

The direction of the normal to the surface is determined using right-hand rule by curling the fingers in the direction of the electric current.

AI!!

=µ B!!!

´= µt

µ!

qqtqt dddWB sin0== dUdWB -= BBU!!×-=-= µqµ cos

^

Torque – generalized equation

1

2

qb a

aq

F

BF

at cosbFnet =qt sinbFnet =

IaBF =

AIdd!!

=µ

ò= AdA!!

ò ´= Bd!!! µt


Magnetic moments

Rn̂

21n̂ 22 RqR

Tq wpµ ==

!AI!!

=µ+µ!

-µ!

n̂2RmIL ww ==!!

Lmq !!

21

=µ

w

x

prwp

2

2dxrdI = 2xA p=

62

2

0

22 LQ

dxxr L wrwpµ ò ==

ò=L

dmxI0

2 ò=L

dxxr0

22rp 2

31ML= L

MQ !!

21

=µ

B!!!

´= µt


Magnetic moment of a spherical shell

dsR

r

TdQdI =AdId =µ

rdsdQ ps 2×=

w

qqps RdRdQ ××= cos2

qswqpµ dRd ×= 34 cos

qqsw RdRdI ×= cos

ò=2/

0

34 cos2p

qqpswµ dR 4

34 Rpsw=

q

2

31 RQw=

µ!

Experimental Physics II - Electric charge in magnetic field 11

Charge moving perpendicular to B

R

B!

v!

BF!

+v!

BF!

+

v!

BF!

+

å = amF !!

^== maqvBFB

mqB

=w

qBmvR =

Cyclotron frequency


Ionization energies

J106.1eV1 19-´»


Arbitrary angle

+x

y

z 0=yaz

y

z

y

R

qBvm

R zx=22zx vv

qBm

+=

ïî

ïí

ì

==

=

aa

sincos

0

0

vvvv

constv

z

x

y

ïî

ïí

ì

==

=

ytvzytvx

tvy y

sincos

0

0


Some examples

Magnetic bottle

Van Allen belts


| | | | | | | | | | | | | | | |

+ + + + + + + + + + + +

Velocity selector

+ v!BF!

EF!

EqBvqFnet!!!!

+´=

qEqvB -=0

BEv =unaffected

m1mm1 -=d

T12=B dBVv =unaffected

m/s1km/s1unaffected -=v


Mass spectrometer

| | | | | | | | | | | | | | | |

+ + + + + + + + + + + +

+ v!BF!

EF!

cqBmvR =

vLqBm c

2=

vsB!

cB!

ELBqBm cvs

2=


Hall effect

Hall voltage

In equilibrium:

HallqEBqvd =

BvE d=Hall

BlvElV d=×=D HallHall

dlnqvI d=d

l

BnqdIV =D Hall d

IBRHº

Hall coefficientSilver: RH = 0.84´10-10 m3/C

+

-


Hall effect thruster


To remember!

Ø Magnetic force acts only upon moving charges.

Ø Magnitude of magnetic field is measured in Tesla.

Ø The direction of this force perpendicular to both the velocity of the particles and to the magnetic field.

Ø The magnitude is proportional to the charge, the speed, the magnetic field and the sine of the angle between v and B.

Ø The same applies to unidirectional conductors, but with charge replaced by the electric current.

Ø For curved conductors, the total force is found by integrating along the conductor.

Ø No net magnetic force acts upon a closed-loop conductor placed in an uniform magnetic field.


To remember!

Ø Magnetic force acting upon a curved conducting wire carrying electric current is found by dividing it on small straight parts and summing up (integrating over) the forces acting upon them.

Ø Homogeneous magnetic filed exert no force on any current loop.

Ø However, it can exert torque.

Ø The toque is maximal when the magnetic filedis parallel to the plane of a flat current loop and zero it is perpendicular to it.

Ø Magnetic moment of a current loop is product ofthe electric current and the loop area.

Ø Any charged rotating object, i.e. having non-zero angular momentum, has as well magnetic moment.


To remember!

Ø If a charged particle enters magnetic field with initial velocity perpendicular to the field lines, it will get trapped by the magnetic field and perform circular motion.

Ø The angular frequency is called cyclotron frequency.

Ø For other angles the trajectory is a helix.

Ø Simultaneous application of electric and magneticfields allows velocity selection of charged molecules.

Ø With one more magnetic field mass spectrometercan be constructed.

Ø In conductors carrying electric current and placedin magnetic field the Hall potential difference isdeveloped due to deflection of moving electrons.

Experimental Physics EP2a · When a charged particle moves parallel to the magnetic field lines,...

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