32.1 MagnetismI. Basics

a. Experimentsb. Understandings

i. Similar to electricity1. Long range2. Two versions3. Attraction/repulsion4. Physical property of matter

c. Compasses and Geomagnetism

32.2 The Discovery of the Magnetic FieldI. Oersted – serendipitous discoveryII. Right-hand rule

III. The Magnetic Field B⃑a. Created at all pointsb. Magnetic field is a vector at each point; magnitude and directionc. Exerts force on magnetic dipoles, parallel to field on North pole, opposite on South pole

IV. Magnetic Field Linesa. Tangent to a field line is in the direction of the magnetic fieldb. The closer the lines, the greater the magnetic field strength

32.3 The Source of Magnetic Field: Moving ChargesI) Moving charges are the source of a magnetic field

A) B⃑point charge=¿hand rule ¿¿B) Biot-Savart LawC) Units = 1 tesla = 1 T ≡ 1 N/A mD) μ0=4 π x10

−7T m/ A – permeability constantE) Thumb is now pointed in the direction of v⃑, and right-hand rule is used to find direction of B⃑

II) SuperpositionA) If there are n point moving charges, then the field will be the sum of all B⃑n ' sB) B⃑total=B⃑1+B⃑2+ B⃑3+⋯+ B⃑n

III) The Vector Cross ProductA) Recalling C⃑ × D⃑=CD sin α, where α is the angle between C⃑ and D⃑

B) Biot-Savart can be written B⃑point charge=μ04 π

q v⃑× r̂r2

32.4 The Magnetic Field of a CurrentI) Current is simply a collection of point charges moving:

a) (∆Q ) v⃑=∆Q ∆ s⃑∆ t

=∆Q∆ t

∆ s⃑=I ∆ s⃑

b) B⃑current segment=μ04 π

I ∆ s⃑× r̂r2

, very short segment of current

II) Example 32.3 and 32.5III) Current Loop

A) B⃑coil center=μ02

¿R

,

i) Nnumber of loopsii) Icurrent in ampsiii) R radius of loops

32.5 Magnetic Dipoles

I) A current loop is a magnetic dipole

II) Magnetic dipole momentA) Define the magnetic dipole moment based on the magnetic field of a loop

(1) Remember the derivation of the electric field of a dipole was predicated by defining the electric dipole moment (a) p⃑=qs ,¿negative ¿ positive

(b) E⃑dipole=1

4 π ϵ 02 p⃑z3

(2) This time we take a look at the magnetic field of a current carrying loop of wire

(a) B⃑loop=μ02

I R2

( z2+R2 )3 /2 , when z≫ R, then ( z2+R2 )3 /2 approaches z3

(b) B⃑loop=μ02I R2

z3=( ππ )Bloop= μ0

4 πI (π R2 )z3

=μ04 π

2 AIz3

(3) μ⃑=AI ,¿ the south pole the north pole

(4) B⃑dipole=μ04 π

2 μ⃑z3

32.6 Ampere’s Law and SolenoidsI) Line integrals

A) Line integral l=∑k∆sk→∫

i

f

ds

B) How long is a line? Find the sum of its tiny little length parts!

C) What if we do a summation of a dot product? ∑kB⃑k ⋅∆ s⃑k→∫

i

f

B⃑ ⋅d s⃑

i) Two cases are important to evaluate, when field lines are parallel to the line and perpendicular to the line

II) Ampere’s Law

A) ∮ B⃑ ⋅d s⃑ circle indicates a line integral where i and f are the same position, a closed path

Situation resembles the path where field always points parallel to the path.

∮ B⃑ ⋅d s⃑=Bl=B (2πd )

Field from a current carrying wire: μ0 I2πd

∮ B⃑ ⋅d s⃑=μ0 I

Three details:1. Independent of shape of curve2. Independent of where current

passes through the curve3. Depends only on the total

amount of current enclosed by the integration curve

III) Magnetic Field of a SolenoidA) Field within the loops is strong and roughly parallel to the central axisB) Field outside is nearly zero

∮ B⃑ ⋅d s⃑=μ0 I

∮ B⃑ ⋅d s⃑=μ0∋¿

Bsolenoid=μ0∋¿l¿

32.7 The Magnetic Force on a Moving ChargeI) Magnetic fields exert Force on current

A) Relationship is dependent on(1) The charge(2) The speed(3) The field strength(4) Angle between v⃑∧B⃑

B) F⃑onq=q v⃑× B⃑ = qvB sinα , direction of right hand rule

(1) Thumb = v⃑(2) Index finger = B⃑(3) Middle finger = F⃑

C) Consequences(1) Only moving charges experience force(2) No force on charges moving parallel or antiparallel(3) Force is perpendicular to both v⃑ and B⃑(4) The force on a negative charge is opposite to v⃑× B⃑(5) For a charge moving perpendicular to the field the force is |q|vB

II) Cyclotron Motion

F=qvB=m v2

r

r=mvqB

&

f= v2πr

→ qB2 πm

Applications:1. The Cyclotron – turning charged particles in particle accelerators2. Van Allen Radiation Belt3. Aurora

III) The Hall Effect

FB=ev dB=Fe=eE=e∆Vw

∆V Hall=w vdB

32.8 Magnetic Forces on Current-Carrying WiresI) Wires carrying current perpendicular to magnetic fields experience force

q=I ∆ t=I lv

F⃑wire=I l⃑ × B⃑=IlB sinα

Direction from right hand rule

II) Force between parallel wires

F ¿wires=I 1l B2=I1 lμ0 I 22 πd

=μ0l I1 I 22 πd

32.9 Forces and Torques on Current Loops

τ=2Fd=2 ( IlB ) ¿

l2=A

¿ μB sinθ

τ⃑= μ⃑× B⃑

I) Electric Motor

32.10 Magnetic Properties of Matter