17 Lorentz Force F ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ v B q x x x x x x x x x x x x x x x x x x v B q → → → → → → → → → → v B q F = 0 F ×
17
Lorentz Force
F
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑v
B
q
x x x x x xx x x x x xx x x x x x
v
B
q
→ → → → →
→ → → → →v
B
qF = 0F
×
Mass spectrometer and v selector
18
19
Magnetic Force on a Current
Force on each charge Force on length ds of wire
€
qv ×B
€
dF = nAdsqv ×B= Ids×B
Torque on a current loop
Null net force
• If plane of loop is not ⊥ to field, there will be a non-zero torque on the loop!
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
B
i
F
F
Force on top path cancels force on bottom path (F = IBL)
Force on right path cancels with force on left one
B
x
.FF
θw
⇒
⇒
magnetic dipole moment: µ = A I
Potential Energy of Magnetic Dipole
When a magnetic dipole is rotated through an angle dθ, the work done is:
dW = -τdθ = -µBsinθdθ dU = -dW= µBsinθdθ Integrating: U = U0 - µBcosθ
• U0 = 0 at position of max torque θ=90deg
B
x
.FFw
θ
µ
⇒
Hall effect
22
Hall Voltage
used to determine sign of charge carriers in a conductor strip
when FE = FB charge carriers no longer move upwards. In the 2 cases of opposite carriers the top part of the strip will be positive or negative (same sign of carriers!). The potential difference sign indicate the sign of carriers. For a metal upper part is negative, for a semiconductor also positive holes
vd=EH/B I = nqvdA => n=IB/(qEHLw)=IB/(qVHL)
L
23
Magnetic field of a moving charge
A single charge in motion:
Permeability of free space:
€
µ0 = 4π ⋅10−7T ⋅m /A
€
B = µ0
4πqv × ˆ r r2
24
Electric current source of magnetic field
Current (flow of electric charges ) in wire produces magnetic field (Oesterd’s experiment)
That magnetic field aligns compass needle
Current
Magnetic field
25
Law of Biot-Sarvart
Each element of current produces a contribution to the magnetic field.
r θI
ds
€
dB =µo
4πIds× ˆ r r2
B out of page
For a single charge in motion
€
B = µ04π
qv ×urr2
For a wire with current
€
qv→ Idl