Lectures 11 & 12: Magnetic Fields and the Motion of Charged Particles

Chapters 26-28 (Tipler)

Electromagnetism

Learning ObjectivesTo introduce the concepts of magnetic fields

•Current

•Magnetic force on a moving charge

•Magnetic field lines

•Magnetic flux

The Nature of Electric Current

Recognizable effects of current flow are:

•heating

•magnetic

•electrolytic

The Nature of Electric Current

Recognizable effects of current flow are:

•heating

•magnetic

•electrolytic

Definition of Current

Suppose a conductor carries a current I

Rate of flow of charge Q past a given cross-section is defined by:

dt

dQI =

The SI unit of current is the ampere: one ampere is defined to be one coulomb per second.

André Marie Ampére (1775-1836)

The Current in a Conductor

ne

Area A

vd

In a time t, volume “swept” out is: Avdt

Charge contained in this volume is:

( )etAvnQ de Δ=Δ

Therefore: eAvnIt

Qde==

The current per unit cross section is called the current density J:

deevnA

IJ ==

Magnetic Field

•A moving charge (current) creates a magnetic field in the surrounding space

•The magnetic field exerts a force Fm on any other moving charge (or current) that is present in the field

The Magnetic Force on a Moving Charge

Experimentally:

A particle of charge +q moving with velocity v in a magnetic field B, experiences a (magnetic) force Fm:

qFm ∝ vFm ∝ θsin ∝mFFm is to v & B

θsinBqvFm =

Fm

θ

B

v

BvqF m ∧=

In vector form

€

rF m = q

r v ×

r B

The unit of B is: N C-1 m-1 s

N C ms-1

This is given a special name tesla (T), in honour of Nikola Tesla

If 1 C of charge moving at 1 m/s perpendicular to a magnetic field experiences a force of 1 Newton, the magnetic field is 1 tesla.

Earth’s magnetic field 5 10-5 TPoles of a large electromagnet 2 TSurface of a neutron star 108 T

Nikola Tesla (1856-1943) a Slovenian born American electrical engineer

Gauss (G) is another unit in

common use: 1G = 10-4 T

In regions where both E and B fields are present, the total force is the vector sum of the electric and magnetic forces:

( )BvEqF ∧+=

in direction of E to v and B

Lorentz Equation

The Earth’s magnetic field at a particular region is represented by

k70sinBj70cosBB OO −=A proton is moving in this magnetic field with a velocity

jsm10v 7 =Obtain an expression for the direction and magnitude of magnetic force acting on the proton.

Worked Exercise

BvqF m ∧=

€

F = q vyˆ j ( )∧ By

ˆ j + Bzˆ k ( )

= qv yByˆ j ∧ ˆ j ( ) + qvyBz

ˆ j ∧ ˆ k ( )

= qvyBzˆ i

= −3 ×10−17 N ˆ i

Magnetic Field Lines

Magnetic Field Lines

NOTE: unlike electric field lines, magnetic field lines are ALWAYS

continuous (no magnetic monopoles) and they do not point in the direction of the force on the moving charge in a magnetic field – they ARE NOT lines of force.

Magnetic Field Lines

•the tangent to a field line at a point P gives the direction of B at that point

•the number of field lines drawn per unit cross sectional area is proportional to the magnitude of B

Magnetic Flux

The magnetic flux B passing through the small area A shown is defined by:

€

φB = Bcosθ × ΔA = B • ΔA

€

φB = B • d Asurface

∫

A

€

φB = B.dAclosed surface

∫ = 0

Gauss’s flux law for magnetism:

€

φE = E .dAclosed surface

∫ = Q

Review and Summary•A magnetic field B is defined in terms of the force Fm acting on a test particle with charge q and moving through the field with velocity v:

€

F m = qv × BThe SI unit for B is the tesla (T)

Review and Summary

•Compare this with the definition of the electric field E

FE = qE

Review and Summary

Gauss’s Law for Magnetism

The net magnetic flux through any closed surface is zero

€

φB = B.d Asurface

∫ = 0

As a result, magnetic field lines always close on themselves

Class Exercise

False

1. True or False: The magnetic force does not accelerate a moving charged particle because the force is perpendicular to the velocity of the particle.

2. What is the force acting on an electron with velocity

in a magnetic field

€

v = 2ˆ i − 3ˆ j ( ) ×106 m/s

€

B = 0.8ˆ i + 0.6 ˆ j − 0.4 ˆ k ( )T

€

F = Fx2 + Fy

2 + Fz2

F = 0.621 pN

1 pN = 10-12 N

€

rF = −0.192ˆ i − 0.128 ˆ j − 0.576 ˆ k pN

Last lecture:Magnetic force on moving charges

€

F m = qv × BMagnetic flux and Gauss’s flux law

€

φB = B.dAclosed surface

∫ = 0

BvqF m ∧=A few special cases:

v parallel to Bv perpendicular to Bv makes an angle θ to B

Today’s lecture:

Magnetic force on a current, torque on a current loop.

BvF ∧=q(a) v parallel to B F = 0

(b) v B F to the plane containing B and v qvBF =

r

mvqvB

2

=

€

ω =2πf = 2π1

T=

qB

m

qB

mvr =€

2πr

v= T

x x x x xx x x x xx x x x x

The direction of B field in sketches

Into the paper,away from you

Out of the paper,towards you

(c) v makes an angle θ with B

(i) a uniform circular motion in which it has the speed vsinθ in a plane perpendicular to the direction of B

(ii) a steady speed of magnitude vcosθ along the direction of B

Helical Motion

B

vθ

The Force on a Current-carrying Conductor

Single Charge BqvF dq =N, the number of charge carriers in volume Al, is: N = nAl

n = charge number density

Total force F:

BnAlqvNFF dq ==( )( ) JAlBlBAnqvF d ==

IlBF =

In General

BlIF ∧=magnetic force on a straight wire segment

The direction of l is defined as the direction of the current I

If the conductor is not straight, consider individual segments and use

BlIdFd ∧=magnetic force on an infinitesimal wire segment

x

x

x

x

B

I

b

F1

The Torque on a Current-Carrying Loop (26-3 Tipler)

l

-F1

F2 -F2

b parallel to yl in the x-z plane,makes an angle to xF1, points in y direction-F1, points in -y,net torque from F1 and -F1 is zero.

The Torque on a Current-Carrying Loop (26-3 Tipler)

φτ sin2 lF ×=φsinlBIb×=φsinBIA=

Fr ∧=τ

O F2

-F2

F2 passes through O, so torque from F2 is zero.

Torque from -F2The direction of the torque??

r

Pointing out towards you

Will the loop accelerate towards you?

The Torque on a Current-Carrying Coil

φτ sin2 lF ×=φsinlBIb×=

φsinBIA=

A

BAI ∧=τ

l

The Torque on a Current-Carrying Coil

BAI ∧=τThis result is true for loops of any shape

Define AI=μ

B∧=μτMagnetic Dipole Moment

The torque tends to align μ and B

€

τ =BIAsinφ

€

τ =BIAsinφ

Unstable equilibrium= 180 degrees

Stable equilibrium= 0

I

Comparison between Magnetic and Electric Dipole Moments

Magnetic Dipole

μB∧=μτ

B

-q

+q

p

Electric Dipole

E

Ep∧=τ

EpU .−=Electric Dipole Potential EnergyMagnetic Dipole Potential Energy

BU .μ−=

Electric dipole,

Molecules behave like electric dipoles

Why do we bother with magnetic dipoles?

v

L

Magnetic Dipole Moment of an Electron in an Atom

2rI πμ ×=

€

I =e

T r

ev

π2=

2

evr=μ

Angular Momentum L = mer x v

€

μ =evr

2

me

me

=e

2me

L

Magnetic Dipole Moment of an Electron in an Atom

Lm

e

e2−=μ

Q.M. L is quantised. Fundamental unit is:

€

L = h

eB m

e

2

h=μ

Fundamental Unit of Magnetic Moment

)T (J mA 10274.9 -1224−×=

Bohr Magneton

All atoms have orbiting electrons,Why only some have a non-zero magnetic moment??

Next lecture:Connecting B to current I