Physics /Electromagnetism /Module 1.4

Applications of Magnetic Forces

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 ( $\vec{F}_E = q\vec{E}$ ) and the magnetic Lorentz force ( $\vec{F}_B = q(\vec{v} \times \vec{B})$ ), 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.

Mass Spectrometer Insight

Carbon isotopes are vaporized and ionized at the source. They enter the chamber with a wide, uncontrolled spread of velocities. Without selection, the ions deflect randomly and crash into the plates. The beam is scattered and blocked.

Net Transverse Force (in Slit)

F_{\text{net}} = \text{N/A}

Isotope 1 Radius (Carbon-12)

r_{12} = \text{N/A}

Isotope 2 Radius (Carbon-14)

r_{14} = \text{N/A}

Selected Beam Velocity (v)

v = \text{N/A}

Interactive 3D canvas allows Orbit rotation and zoom.

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:

q \textcolor{#EAB308}{E} = q \textcolor{#10B981}{v} \textcolor{#8B5CF6}{B}

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 (F_E = qE) must perfectly balance the magnetic Lorentz force (F_B = 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.

Cyclotron Accelerator Insight

A positive proton is injected at rest at the very center of the gap between the two dees. It remains stationary, waiting for an electric field to provide its initial acceleration.

RF Voltage Frequency (f)

f = \text{constant } f = 22.9\text{ MHz}

Gap Electric Field (E)

E = 0\text{ kV/m}

Proton Velocity (v)

v = 0.00\text{ m/s}

Current Orbit Radius (r)

r = 0.00\text{ cm}

Interactive 3D canvas allows Orbit rotation and zoom.

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:

q \textcolor{#10B981}{v} \textcolor{#8B5CF6}{B} = \frac{m \textcolor{#10B981}{v}^2}{\textcolor{#E07A5F}{r}}

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 (F_B = qvB) that acts centripetally to keep it in a circular path of radius r. We set the magnetic force equal to the centripetal force: F_B = F_c.

Concept Checks

Mass Spectrometer Calculation

An unknown ion of charge

+e

enters a mass spectrometer with selector fields

E = 1.0 \times 10^5\text{ V/m}

and

B = 0.20\text{ T}

. In the deflection chamber, the magnetic field is also

B_0 = 0.20\text{ T}

. If the ion strikes the detector strip at a radius

r = 0.026\text{ m}

, what is the mass of the ion? (Charge $e = 1.60 \times 10^{-19}\text{ C}$ )

Cyclotron Sync Frequency

A cyclotron with a uniform magnetic field of

1.5\text{ T}

is designed to accelerate protons. What must be the alternating voltage frequency

f

required to keep the protons in sync? (Mass of proton $m = 1.67 \times 10^{-27}\text{ kg}$ , Charge $q = 1.60 \times 10^{-19}\text{ C}$ )