Variables
Strategy
- Express the change in the kinetic energy of the ion during the acceleration
phase in terms of its mass and quantities that can be controlled or measured by
the experimenter. By “acceleration phase,” we mean the linear acceleration in the
electric field. - Express the change in the potential energy of the ion during the acceleration
phase in terms of some of the same quantities, including the potential difference
between the plates. - Use the conservation of energy to relate kinetic energy to potential energy, and
solve for the unknown mass of the charged particle.
Physics principles and equations
We use the definition of kinetic energy.
KE = ½ mv^2
The radius of the circular path of a charged particle moving perpendicular to the
magnetic field is
r = mv/qB
The potential difference equals the change in PE per unit charge.
ǻV = ǻPE / q
The principle of the conservation of energy applied to mechanical energy states that
ǻKE + ǻPE = 0
Since the magnetic field changes the particle’s direction but not its speed, its KE is not changed by the magnetic field.
Step-by-step derivation
In the first stage of the derivation we write ǻKE in terms of the mass m and the speed v of an ion as it leaves the electric field and enters the
magnetic field. Neither m nor v can be directly observed. We use another equation to replace vby quantities that can be controlled or observed
in the laboratory.
m = mass, q = charge
r = radius
B = magnetic field strength
ǻV = potential difference
This mass spectrometer is
testing a hydrogen molecule
(H 2 +) having charge +e. What is
its mass?
q = 1.6×10í^19 C
m = 3.3×10í^27 kg
change in kinetic energy ǻKE
mass of ion m
accelerated speed of ion v
radius of circle in magnetic field r
charge of positive ion q = +1.6×10í^19 C
magnetic field strength B
change in potential energy ǻPE
potential difference across plates ǻV