PHYS 208 Lecture 14

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Magnetic Fields (cont'd)

Magnetic force on moving electric charge:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = q\vec{v} \times \vec{B}}

Velocity Selector

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \tfrac{1000}{.1}\ \mbox{V/M}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = .2\ \mbox{T}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} qE &= qvB \\ E &= vB \\ v &= \frac{E}{B} \end{align}}

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{qBR}{v} = \frac{qB^2R}{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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{1/2} = \frac{\mathrm{distance}}{\mathrm{velocity}} &= \frac{\pi R}{v} \\ &= \frac{\pi R}{\frac{q}{m} BR} \\ &= \frac{m \pi}{q B} \end{align}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \frac{2 \pi m}{q B}}

Magnetic Force on Current

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = I\vec{\ell} \times \vec{B}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = I \int \mathrm{d}\vec{\ell} \times \vec{B}}

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