PHYS 208 Lecture 14

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Magnetic Fields (cont'd)

Magnetic force on moving electric charge:

${\vec {F}}=q{\vec {v}}\times {\vec {B}}$

Velocity Selector

Suppose $E={\tfrac {1000}{.1}}\ {\mbox{V/M}}$ and $B=.2\ {\mbox{T}}$:

${\begin{aligned}qE&=qvB\\E&=vB\\v&={\frac {E}{B}}\end{aligned}}$

So $v={\frac {1000/.1}{.2}}\ {\mbox{m/s}}$

Other Cool Physics Gadgets

Mass Spectrometer

Inner workings of a Mass Spectrometer

Mass spectrometers accelerate charged particles, use a velocity selector to achieve a known velocity, and apply a uniform magnetic field to create a "mass selector" that bends the path of the particles onto a detector plane. Mass can be measured using the following equation:

$m={\frac {qBR}{v}}={\frac {qB^{2}R}{E}}$

Cyclotron

Two D-shaped pita-pocket electrodes connected to an alternating voltage source in a vacuum. Particles get a kick into the oppositely charged electrode, and in region of uniform magnetic field, it bends around the D, then the voltage oscillates one tick. This acceleration process happens until the physicist is ready for particle extraction. The time to take one half-revolution is

${\begin{aligned}t_{1/2}={\frac {\mathrm {distance} }{\mathrm {velocity} }}&={\frac {\pi R}{v}}\\&={\frac {\pi R}{{\frac {q}{m}}BR}}\\&={\frac {m\pi }{qB}}\end{aligned}}$

Notice that the time does not depend on the radius of the cycloton electrodes. Thus the cyclotron frequency of a revolution is given by

$t={\frac {2\pi m}{qB}}$

Magnetic Force on Current

Magnetic field applies a force to the charged particles inside a wire carrying electric current:

${\vec {F}}=I{\vec {\ell }}\times {\vec {B}}$

For a wire that bends through a potentially non-uniform magnetic field:

${\vec {F}}=I\int \mathrm {d} {\vec {\ell }}\times {\vec {B}}$

This can get nasty very quickly, so we'll keep our geometries simple...