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

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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:

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µ0 = 4π ⋅10−7T ⋅m /A

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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

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dB =µo

4πIds× ˆ r r2

B out of page

For a single charge in motion

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B = µ04π

qv ×urr2

For a wire with current

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qv→ Idl