Introduction

Daniel FitzGreen

Charge to Mass Ratio

Introduction

The charge-to-mass ratio $(e/m)$ of the electron, or any other charged particle, is a value that comes up repeatedly in any branch of physics that is interested in elementary particles. The value is significant because it is often easily measurable, whereas measuring the electron mass exclusively is exceedingly difficult. Determinations of the electron mass are made using the charge-to-mass ratio and theoretical considerations for the value of the charge.

When a charged particle such as an electron moves in a magnetic field, in a direction at right angles to the field, it is acted on by a force given by

(1) $\begin{equation*} F = Bev, \end{equation*}$

where $B$ is the magnetic flux density in Tesla, $e$ is the charge on the electron in Coulombs, and $v$ is the velocity of the electron in m/s. This force, called the Lorentz force, causes the particle to move in a circle in a plane perpendicular to the magnetic field. The radius of this circle is such that the force exerted on the particle by the magnetic field furnishes the required centripetal force. Therefore,

(2) $\begin{equation*} \cfrac{mv^2}{r}=Bev,\end{equation*}$

where $m$ is the mass of electron in kg, and $r$ is the radius of the circle in m. Presuming we know the charge of the electron, the charge-to-mass ratio can be solved if we know the magnetic field and the velocity of the electron.

If the velocity of the electron is due to its being accelerated through a potential difference $V$ , it has a kinetic energy of

(3) $\begin{equation*} \cfrac{1}{2} mv^2 = eV \end{equation*}$

where $V$ is the accelerating potential of the apparatus in volts, and $e$ is the elementary charge in C.

The magnetic field is produced using a pair of solenoids in a Helmholtz configuration, the value of which is given by

(4) $\begin{equation*}B = \cfrac{8\mu_0NI}{a\sqrt{125}},\end{equation*}$

where $I$ is the current through the coils in A, $a$ is the mean radius of the coils, in m, and $\mu_o$ is the magnetic permeability of free space, which has the CODATA value of 1.25663706212 (19) 10⁻⁶ N/A². You probably won’t need all of those digits.

With the above information, $e/m$ can be computed, producing a value in units of C/kg.

In this experiment, students will create an electron beam by accelerating electrons from a heated filament, and change its trajectory by applying current to the Helmholtz coils that surround an electron beam, thereby changing the magnetic field.

Introduction

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