Discover how combining electric and magnetic fields allows us to manipulate, filter, and accelerate charged matter at the atomic scale.
When electric and magnetic fields are crossed or sequenced, they become powerful tools for
manipulating matter. By carefully balancing the electrostatic force (FE=qE) and the magnetic Lorentz force (FB=q(v×B)), we can sort isotopes by their mass or accelerate subatomic particles to near-light speeds
in a compact laboratory space.
The Mass Spectrometer
A mass spectrometer is an analytical instrument that separates
charged atoms or isotopes based on their mass-to-charge ratio (m/q).
This process occurs in three main stages:
Ionization: The sample is vaporized and bombarded with electrons to create positive
ions.
Velocity Selection: The ions are injected into a region containing crossed electric
and magnetic fields, which filters out all but a single, uniform velocity beam.
Magnetic Deflection: The uniform-velocity beam enters a uniform magnetic field,
curving the ions into semicircular paths whose radii depend directly on their mass.
Derivation of the Mass Equation
How do we determine the exact mathematical mass of an unknown isotope? By equating the
electric and magnetic forces acting on the ion at each stage, we can derive the formula
step-by-step.
Click through the visual steps below to watch the algebraic substitution unfold and discover
the final mass spectrometer equation:
qE=qvB
1. Velocity Selector Balance
Equating the electric and magnetic forces
Within the velocity selector, crossed electric and magnetic fields pull the ion in opposite directions. For the ion to travel in a straight line, the electrostatic force (FE = qE) must perfectly balance the magnetic Lorentz force (FB = qvB).
The Cyclotron
Before the advent of modern synchrotron loops, the cyclotron—invented by Ernest O. Lawrence in 1929—was the premier particle accelerator. Accelerating
particles in a straight line requires a miles-long tunnel to achieve high speeds. The
cyclotron solves this by using a magnetic field to loop particles in circles, allowing them to
pass through the same accelerating electric gap thousands of times.
The cyclotron consists of two hollow, D-shaped copper electrodes called "dees", placed in a strong perpendicular magnetic field. The dees act as Faraday cages: inside
them, the electric field is zero, so the B-field simply rotates the particles in a semicircle.
Between the dees lies a narrow gap where a high-frequency alternating electric field provides
a rapid push every time the particle crosses.
Derivation of the Cyclotron Frequency
To continuously accelerate the particle, the electric field across the gap must flip direction
every time the particle exits a dee. This requires an alternating current (AC) voltage source
synchronized perfectly with the particle's motion.
Click through the visual steps below to watch the algebraic cancellation unfold and discover
why the orbital frequency is completely independent of the proton's velocity:
qvB=rmv2
1. Centripetal Force Balance
Equating magnetic and centripetal force inside a dee
Inside the copper dees, the electric field is zero. Only the uniform magnetic field (B) acts on the proton, providing a magnetic Lorentz force (FB = qvB) that acts centripetally to keep it in a circular path of radius r. We set the magnetic force equal to the centripetal force: FB = Fc.
Concept Checks
1
Mass Spectrometer Calculation
An unknown ion of charge +e enters a mass spectrometer with selector fields E=1.0×105 V/m and B=0.20 T. In the deflection chamber, the magnetic field is also B0=0.20 T. If the ion strikes the detector strip at a radius r=0.026 m, what is the mass of the ion? (Charge e=1.60×10−19 C)
2
Cyclotron Sync Frequency
A cyclotron with a uniform magnetic field of 1.5 T is designed
to accelerate protons. What must be the alternating voltage frequency f required to keep the protons in sync? (Mass of proton m=1.67×10−27 kg, Charge q=1.60×10−19 C)