Helmholtz

Finding the Charge to Mass Ratio of the Electron Via a
Uniform Magnetic Field Generated by Helmholtz Coils

Abstract

The charge to mass ratio of the electron was found to be [pic] with a percent difference from the accepted value ([pic]) of 1.1% when adjusted for the Earth’s magnetic field. Without taking the Earth’s magnetic field into consideration, the charge to mass ratio was found to be [pic] with a percent difference of 5.3%.

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MOORPARK COLLEGE
THEORY

We seek the magnetic field along the axis of the two Helmholtz coils in figure 1.

[pic]
Figure 1

First, considering the magnetic field due to the coil on the left, the magnetic field, B, along the normal axis of a coil is defined as,

[pic] (1)

Where N is the number of loops in the coil.

By definition, the magnetic dipole moment can be expressed as,

[pic] (2)

Furthermore, substituting (2) into (1) yields,

[pic]

By extrapolating this result, the magnetic field due to the coil on the right is,

[pic]

[pic]

Since the current in the two coils flow in the same direction, then the net magnetic field will be the sum of the two corresponding magnetic fields.

[pic] (3)

For a point midway between the coils,

[pic]

therefore,

[pic]

[pic] (4)

Next, substituting for the magnetic dipole moment in equation (2),

[pic]

[pic] (5)

The force on an electron within this magnetic field is,

[pic] (6)

where e is the charge on an electron, v is the velocity, and r is the radius of the electron path. Solving for velocity in (6),

[pic] (7)

Additionally, potential energy can be related to potential difference as follows,

[pic] (8)

where V is