Physics307L:People/Knockel/Notebook/070926

Electron's e/m Ratio

Experimentalists: Nik, where are you?

Goal

If a beam of electrons moving at a certain speed is bent by a magnetic field, the entire beam will curve into a circle of fixed radius. This result means that there is a charge to mass ratio for the electron, and it can be calculated! The accepted value for the charge to mass ratio of an electron is [math]\displaystyle{ \frac{e}{m}=1.759\times10^{11} \frac{C}{kg} }[/math].

Equipment

All the equipment setup correctly.

• 2 weak direct current power supplies (10V and 2A are the max outputs needed)
• 1 strong direct current power supply (500V needed)
• 2 multimeters
• banana plugs
• The Device!
  • Helmholtz coil
  • Vacuum tube with helium gas and electron gun inside
  • Mirror with ticks for measuring length
  • Black cloth hood to block out light

Setup

To plug everything in I...

• acquired The Device!
• plugged in the three power supplies into the wall and the two multimeters
• connected one of the weak power supplies into the heater for the electron gun with banana plugs and turned it on not exceeding 6.3V
• plugged in the other weak power supply into the Helmholtz coil using banana plugs, but intercepted the circuit to connect a multimeter in series to measure current, and then turned on the power to the Helmholtz coil and multimeter so there were about 2 amps flowing
• plugged in the large power supply to power the electrodes of the electron gun using banana plugs, and turned it on so that about 300V are applied
• connected a multimeter to The Device! to measure voltage across electrodes

To adjust the electron beam I...

1. turned off lights
2. waited for heater to heat up (after about two minutes it is at equilibrium)
3. adjusted voltage to electrodes and current to Helmholtz coils so that the beam made a helix
4. rotated the vacuum tube so the helix became a circle

My Procedure

I sat on my ass and took some sample data. When I do the real procedure I'll tell about it here

Theory

I will find two expressions for velocity of the electron beam and then set them equal to each other.

First expression for velocity

If the beam is moving in a circle, the velocity, [math]\displaystyle{ v }[/math], can be discovered in terms of magnetic field strength (B), charge (e), radius of circle (r), and mass of electron (m). The force (radial) required to keep anything in circular motion is [math]\displaystyle{ F_r=-\frac{m v^2}{r} }[/math] The force (radial) from a magnetic field is [math]\displaystyle{ F_r=-e v B }[/math] Solving for v in these two equations by first setting [math]\displaystyle{ F_r=F_r }[/math] gives [math]\displaystyle{ v=\frac{reB}{m} }[/math].

Second expression for velocity

The electrons will be accelerated between two plates. The voltage between these plates, [math]\displaystyle{ V }[/math], is the only significant means for the electron to speed up since the electric field practically goes to zero on the other side of the plate and since gravity is negligible with the speeds we are dealing with. Therefor, kinetic energy (KE) equals the negative of the change in potential energy ([math]\displaystyle{ \Delta }[/math]PE) between the two plates, which equals charge (e) times the voltage between the plates (V)... [math]\displaystyle{ KE=\frac{1}{2}m v^2=-\Delta PE=Ve }[/math] Solving for [math]\displaystyle{ v }[/math] gives [math]\displaystyle{ v=\sqrt{\frac{2Ve}{m}} }[/math]

Setting both expression equal to each other and solving for e/m

[math]\displaystyle{ \frac{e}{m}=\frac{2V}{B^2 r^2} }[/math]

Data

A datum is good but data are better.

Where to mail the Nobel Prize money

Bradley Knockel
My street address
Albuquerque NM 87106