doPhysics · Electromagnetism 2
Magnetic Force on Moving Charges
The force a magnetic field exerts on a moving charge, qv × B, and the paths it produces: circles, helices and the guided spirals of the aurora. Crossed fields that select one speed, Thomson's e/m and the mass spectrometer, the cyclotron, and the Hall effect.
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Why this chapter
Why this chapter exists
A magnetic field ignores a charge at rest and pushes a moving one sideways, never along its path. That one rule, F = qv × B, bends the beam of a picture tube. It steers the particles of the solar wind into the polar sky and holds the protons of a cyclotron on their spiral. The force never does work, so a charge in a field keeps its speed and simply turns. Its circle takes the same time at every speed. This chapter follows the rule from a single charge to three kinds of machine. They weigh ions, accelerate them, and reveal the sign of the charges inside a metal.
You will learn
By the end of this chapter you will
- Write the magnetic force on a moving charge, F = qv × B, find its size and direction, and explain why it does no work.
- Find the radius, period and frequency of a charge circling in a uniform field, and the arc it turns through in a strip of field.
- Resolve a velocity along and across the field and find the radius, pitch and period of the resulting helix.
- Explain how the Earth's field guides solar particles to the poles and lights the aurora.
- Use crossed fields to select one speed, \(v = E/B\), and follow Thomson's measurement of \(e/m\) and the mass spectrometer.
- Explain the resonance condition of the cyclotron, find the energy of its beam, and say why it cannot accelerate electrons far.
- Derive the Hall voltage IB/(nqd) and use its sign to find the sign of the charge carriers.
Before you begin
Before you begin
From the chapters Dynamics of Circular Motion and Torque and Equilibrium you should know the centripetal force \(mv^2/r\) and the cross product of two vectors. You should know the work-energy theorem and the power \(F \cdot v\) from the chapter Work, Kinetic Energy and Power. From the chapters on electrostatics you should know that an electric field pushes a charge with \(qE\). It gives the charge an energy \(qV\) across a potential difference. You should also know the drift velocity of the charges in a current.
Lessons
1
A Force That Needs Motion
The magnetic field and the force it exerts on a moving charge, F = qv × B. Its size, its direction by the right-hand rule, and the one thing it can never do, which is work.
Magnetic field and the lorentz forceDirection by the right hand ruleMagnetic force does no work
2
Circles in a Field
A charge moving across a uniform field turns in a circle of radius \(mv/qB\), and its period \(2\pi m/qB\) does not depend on the speed. The radius from momentum or energy, and the part of a circle traced through a strip of field.
Circular motion in a uniform fieldPeriod independent of speedRadius from energy or momentumEntering and leaving a field region
3
Helices and the Aurora
A charge whose velocity is at an angle to the field drifts along it while it circles round it, and traces a helix. Its radius, pitch and period, and how the Earth's field lines guide solar particles into the polar sky.
Resolving the velocity along and across bRadius pitch and period of a helixPitch equals circumference at 45 degreesCharged particles guided by field lines
4
Crossed Fields: the Velocity Selector and Thomson's e/m
An electric and a magnetic field at right angles let only one speed, E/B, through undeflected. Thomson used that balance to measure e/m for the electron, and the mass spectrometer uses it to weigh ions and separate isotopes.
Crossed fields and the velocity selectorThomsons measurement of e by mThe mass spectrometer
5
The Cyclotron
Two hollow dees in a magnetic field and an alternating voltage across their gap give a charge many small kicks. The half-turn time does not grow with the radius, so one oscillator frequency stays in step; the beam leaves with \(K = q^2 B^2 R^2/2m\).
How the cyclotron acceleratesThe resonance conditionMaximum energy of the beamWhy a cyclotron cannot take electrons far
6
The Hall Effect
A current in a strip placed in a magnetic field pushes its moving charges to one face, until a sideways electric field balances them. The Hall voltage IB/ned, the sign of the carriers it reveals, and the Hall sensor.
Charge pushed sideways in a currentThe hall voltageThe sign of the carriers and the hall sensor
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- aurora
- The glow of the upper atmosphere near the poles. It is lit by charged particles from the Sun that spiral down the Earth's field lines and strike the air.
- charge carriers
- The free charges whose drift makes up an electric current: electrons in a metal, and positive or negative carriers in other materials.
- crossed fields
- An electric field and a magnetic field at right angles to each other and to the velocity of a charge. They are arranged so that their forces on it oppose each other.
- cyclotron
- An accelerator in which ions turn half circles inside two hollow dees in a magnetic field. They gain energy at every crossing of the gap between the dees.
- cyclotron frequency
- The frequency \(f = qB/2\pi m\) of a charge circling in a uniform magnetic field. It does not depend on the speed of the charge or the radius of its circle.
- dees
- The two hollow, flat, D-shaped electrodes of a cyclotron. Inside them the ions are shielded from the electric field and only the magnetic field acts.
- Hall effect
- The appearance of a voltage across a current-carrying conductor placed in a magnetic field, because the field pushes the drifting carriers to one side.
- Hall voltage
- The voltage across the width of a current-carrying strip in a magnetic field, \(V_H = v_dBw = IB/(nqd)\). Its sign reveals the sign of the charge carriers.
- helix
- The spring-shaped path of a charge whose velocity makes an angle with a uniform field: a circle across the field carried steadily along it.
- isotope
- One of the forms of an element with the same number of protons, and so the same chemistry, but a different mass. Neon-20 and neon-22 are isotopes of neon.
- Lorentz force
- The force on a charge \(q\) in electric and magnetic fields, \(\vec F = q(\vec E + \vec v\times\vec B)\). Its magnetic part, \(q\,\vec v\times\vec B\), acts only on a moving charge.
- magnetic field
- The field, set up by moving charges and currents, that pushes on other moving charges. Its strength \(B\) is measured in tesla; crosses mark a field into the page and dots a field out of it.
- mass spectrometer
- An instrument that sends ions of one speed into a magnetic field and sorts them by the radius of their half circles, measuring \(m/q = rB_0B/E\).
- pitch
- The distance a charge on a helix advances along the field in one turn, \(p = v\cos\theta\,T = 2\pi mv\cos\theta/qB\).
- resonance condition
- The requirement that a cyclotron's oscillator run at the ions' frequency of revolution, \(f_{osc} = qB/2\pi m\), so that the dees reverse once every half circle.
- right-hand rule
- A way to find \(\vec v\times\vec B\): fingers of the right hand along \(\vec v\), curled towards \(\vec B\); the thumb points along the product. The force on a negative charge points the other way.
- specific charge
- The ratio of the charge of a particle to its mass, \(q/m\). For the electron \(e/m = 1.76\times10^{11}\,\text{C/kg}\).
- tesla
- The SI unit of magnetic field, \(1\,\text{T} = 1\,\text{N/(A m)}\). It is the field that pushes a charge of 1 C, moving at 1 m/s across it, with a force of 1 N.
- velocity selector
- A pair of crossed fields with an exit slit that lets through undeflected only the charges moving at \(v = E/B\), whatever their charge or mass.