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%.
STUDENT:
PARTNER:
INSTRUCTOR:
DATE:
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
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%.
STUDENT:
PARTNER:
INSTRUCTOR:
DATE:
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