Magnetism 1

now for vc γ ≈ 1
⎛ q ⎛−µ 0 ⎞ qv⎞
→Fy≈ q⎜ + v⋅
2 ⎜ ⎟⋅
2 ⎟
⎝4πε 0r
⎝ 4π
⎠ r ⎠
Eel v⋅B
in vector form: v = v ⋅xˆ
2 ⎡ µ 0qv
⎤
= q⎢Eyˆ− ( xˆ zˆ)
2
4π
r
− × ⎥
⎢⎣ yˆ
⎥⎦
(
µ
)
0qv
= q E+ v× B , where Bz=
2
4π
r
→ F = q⋅ (
v× B)
mag
Classical picture
Magnetic moments originate from circular electric currents.
1 ⎛ r ⎞
Biot-Savat: δH = ⋅I⋅ δs
2 ⎜ × ⎟
4π
r ⎝ r ⎠
e
ds
magnetic moment originates from a circular current loop
dµ = I⋅d
S
µ =
dS =
magnetic moment
surface enclosed by the current
2
µ = I dA = 4πr
⋅I
∫
for a circular loop
r
4π
e 2
= r ;
τ
e
I =
τ
for the case of an electron circulating around a ring with period τ
This can be related to the angular momentum of the electron.
le = r× p = r× ve⋅ me 2π
r
where v=
τ
2
2π
r
= me
τ
e
⇒ µ =− ⋅le 2me
relation between angular momentum and magnetic moment of a
circulating electron
They are antiparallel because of the negative charge of the electron.
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