Electrons in fields

The specific charge of a particle is its charge-to-mass ratio (e/m). For electrons this value is very large, which was key evidence that electrons are much lighter than ions such as H⁺. We will look at two methods that were used to find the specific charge of electrons.

The Setup

An electron gun fires electrons into a region of uniform magnetic field , produced by coils. The beam follows a circular path of radius , which can be measured directly on the tube.

Why the beam is visible: The tube contains low-pressure gas. Electrons collide with gas atoms causing excitation; when atoms de-excite they emit photons, making the beam path glow. The diagram below shows the same set up with all the important quantities that need to be measured.

Note the filament voltage is simply to heat the filament inside the electron gun until it can release electrons. The anode voltage is used to make the electric field which accelerates the electron inside the electron gun.

Key measurements:

• anode voltage, (accelerating voltage of the electron gun)

• magnetic flux density, (from coil current / calibration)

• radius of the electron beam,

Deriving the fine beam tube equation for e/m

Energy (acceleration):

Circular motion (magnetic force provides centripetal force):

Rearrange to eliminate v:

Substitute into to get:

Thomson’s comparison with H⁺ ions (what could he conclude?)

Thomson found for cathode ray particles was about larger than the known specific charge of H⁺.

From the numbers alone he could conclude at least one of these must be true:

• the particles have a much smaller mass than H⁺, OR

• the particles carry a much larger charge than H⁺

The later accepted conclusion: a new fundamental particle exists (the electron), with mass far smaller than a proton.

Electrons first pass through a velocity selector (crossed electric and magnetic fields) to select one speed, then they pass into a separate electric field region where they are deflected.

Setup 1:

Velocity selector: selecting a known speed

Variables (selector region):

• electric field strength in the selector

• magnetic flux density in the selector

• pd across selector plates

• separation of selector plates

• electron speed

Why electrons are undeflected (brief):

The electric and magnetic forces act in opposite directions. For the selected speed, these forces are equal, so the resultant force is zero and the beam goes straight.

Condition for no deflection:

Set-up 2: using deflection in electric field to find :

Variables (deflection plates region):

• electric field strength between deflection plates

• pd across deflection plates

• separation of deflection plates

• length of deflection plates

• vertical deflection (measured at end of plates)

• time spent between plates

• vertical acceleration in the plates

Derivation:

First, find the electric field strength between the plates:

The electric field causes the electron to accelerate, so:

The time spent between the plates is:

The vertical motion gives:

Substitute into the specific charge equation:

Substitute :

J.J.Thomspson compared the specify charge of a H+ ion to the one calculated in these experiments. He concluded that a new fundamental particle ( i.e. the electron) exists which either has a much smaller mass or much greater charge.

Measuring showed cathode ray particles are identical regardless of the gas or electrodes. This helped establish the electron as a fundamental particle and changed the model of the atom