Precision & Power

AP Physics C: Unit 7.3

Topic: Mass Spectrometry and Magnetic Torque

Theoretical Core

"Magnetic fields allow us to measure the invisible (mass of atoms) and generate the mechanical (torque in motors)."

Primary Mechanics

Velocity Selection

\( v = E / B \)

Mass Separation

\( m = qBR / v \)

Rotational Work

\( \tau = \mu \times B \)

I. Precision Engineering with B-Fields

The Lorentz force is not merely a theoretical construct; it is the fundamental engine behind some of modern science's most critical tools. In this blueprint, we explore two pinnacle applications: the Mass Spectrometer, which utilizes deflection to identify chemical isotopes, and the Current Loop, which converts electrical energy into mechanical torque.

Crossing the Fields

Before we can measure mass, we must first control velocity. Most particle sources produce a spectrum of speeds. To isolate a specific velocity, we use a Velocity Selector—a region where electric (\(\vec{E}\)) and magnetic (\(\vec{B}\)) fields are crossed perpendicularly.

Diagram 1.1: The Crossed-Field Velocity Selector

+++

---

+q

\(F_B = qvB\)

\(F_E = qE\)

For a particle to travel in a straight line, the net force must be zero (\(\sum F = 0\)). This implies that the magnetic force (upward via RHR-2 for \(\vec{B}\) out) must exactly cancel the electric force (downward).

II. Anatomy of the Mass Spectrometer

Once a beam of ions has been "filtered" to a specific velocity \(v\), it enters the second stage: the Magnetic Deflection Chamber. Here, the electric field is removed, and only a uniform magnetic field \(\vec{B}_0\) remains.

The Deflection Physics

"Inside the chamber, the magnetic force acts as a centripetal force. Since all ions have the same charge and same velocity, their separation depends solely on their mass."

\( m = \frac{q B_0 R}{v} \)

Detectors placed at different radii \(R\) identify the abundance of specific isotopes. Heavier isotopes trace a larger radius, while lighter ones curve more sharply.

Diagram 2.1: Isotope Separation

Ion Entry

Light Isotope (R1)

Heavy Isotope (R2)

Step-by-Step Measurement

Ionize: Sample is turned into positive ions.
Select Velocity: Ions pass through crossed \(E\) and \(B\) fields. Only those with \(v = E/B\) emerge.
Deflect: Ions enter uniform \(B_0\). They follow circular arcs of radius \(R\).
Detect: Position of impact gives \(R\), allowing for the calculation of mass \(m\).

III. Torque on a Current Loop

Moving from point charges to macroscopic systems, we consider a rectangular wire loop of width \(a\) and height \(b\), carrying a current \(I\) in a uniform magnetic field \(\vec{B}\). This system is the precursor to the electric motor.

Diagram 3.1: The Force Distribution

Axis

F1

F2

"Field \(\vec{B}\) points right. Forces on top/bottom cancel. Forces on vertical sides create a couple that rotates the loop."

Analyzing the Force Pairs

Vertical Sides (Height \(b\))

Side 1 carries current \(I\) up. Side 2 carries current \(I\) down. Since both are perpendicular to \(\vec{B}\), they feel forces of magnitude \(F = IbB\). These forces point in opposite directions, creating torque about the central axis.

Horizontal Sides (Width \(a\))

The forces on these sides are parallel to the axis of rotation. While they exert tension on the wire, they contribute zero torque because their line of action passes through the axis.

IV. The General Torque Formula

To find the total torque, we look at the loop from the top down (the side view of width \(a\)). Let \(\theta\) be the angle between the magnetic field \(\vec{B}\) and the area vector \(\vec{A}\) (perpendicular to the loop's surface).

THE DERIVATION

\(\tau = F_1 \frac{a}{2} \sin\theta + F_2 \frac{a}{2} \sin\theta\)

\(\tau = (IbB) a \sin\theta\)

\(\tau = I A B \sin\theta\)

Where \(A = ab\) (the area of the loop).

Geometric Interpretation

The torque depends on the orientation of the loop relative to the field:

• Maximum Torque (\(\theta=90^\circ\)): When the loop is parallel to the field, the moment arm is maximized.
• Zero Torque (\(\theta=0^\circ\)): When the loop is perpendicular to the field, the forces pull outward, but do not rotate.

Area Vector \(\vec{A}\)

Top View: Side "a"

V. The Magnetic Dipole Moment (\(\mu\))

In AP Physics C, we generalize the torque formula using a new vector quantity called the Magnetic Dipole Moment. This allows us to treat any current loop—regardless of shape—using a single property.

Definition of \(\mu\)

\[ \vec{\mu} = N I \vec{A} \]

Where \(N\) is the number of turns in the coil, \(I\) is current, and \(\vec{A}\) is the area vector (direction defined by RHR-1).

General Torque Formula

\[ \vec{\tau} = \vec{\mu} \times \vec{B} \]

The cross product naturally accounts for the \(\sin\theta\) relationship. The torque always acts to align \(\vec{\mu}\) with \(\vec{B}\).

Magnetic Potential Energy

Just like an electric dipole in an electric field, a magnetic dipole has potential energy based on its orientation:

\[ U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\theta \]

Lowest Energy (\(\theta=0^\circ\)): \(\mu\) is aligned with \(B\). Stable equilibrium.
Highest Energy (\(\theta=180^\circ\)): \(\mu\) is anti-aligned. Unstable equilibrium.

Precison & Power Summary

MASS SPEC:

Separates by mass using \(m = qBR/v\). Heavier = Larger Radius.

VELOCITY SELECTOR:

Filters ions using \(v = E/B\). Forces must balance.

MAGNETIC TORQUE:

Causes rotation \(\tau = \mu \times B\). Basis for all DC motors.

© 2026 AP PHYSICS MASTERY Applications Blueprint Article Page 05 of 05

Magnetic Applications Blueprint Article

Magnetic Applications Blueprint Article

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