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Magnetic Fields

Introduction

A magnetic field is a vector field that describes magnetic influence in space. In school and introductory university physics, the symbol $𝐁$ is used for the magnetic flux density, commonly called the magnetic field. Its SI unit is the tesla (T). Magnetic fields are produced by moving electric charges, electric currents, magnetized materials, and changing electric fields. They affect moving charges, current-carrying conductors, and magnetic dipoles.

You encounter magnetic fields in compasses, loudspeakers, electric motors, generators, particle accelerators, magnetic sensors, data-storage systems, and medical imaging. At Grades 11–13, the central challenge is not only to remember formulas but to connect field geometry, vector direction, force, current, and magnetic flux.

The field-line picture of a bar magnet is a model of a vector field. Outside the magnet, the arrows run from the magnetic north pole toward the magnetic south pole; inside the magnet they continue back, forming closed loops. Field lines are not physical strings. They visualize the direction of $𝐁$ , while their spacing gives a qualitative indication of field strength.

Learning Objectives

By the end of this aiMOOC, you should be able to:

Magnetic field: Describe a magnetic field as a vector field and interpret field-line diagrams.
Lorentz force: Determine the magnitude and direction of the magnetic force on a moving charge.
Electric current: Explain how currents create magnetic fields and apply right-hand rules.
Biot–Savart law: Use the law conceptually and quantitatively for simple current geometries.
Ampère's law: Apply symmetry to long straight conductors and idealized solenoids.
Magnetic flux: Calculate flux through a surface and explain Gauss's law for magnetism.
Magnetic dipole moment: Analyze forces and torques on current loops and magnetic dipoles.
Magnetic materials: Compare ferromagnetic, paramagnetic, and diamagnetic responses.
Geomagnetism: Connect magnetic-field models to Earth's field and technological applications.

Foundations: Fields, Vectors, and Field Lines

What a Magnetic Field Represents

At every point in space, $𝐁$ has both a magnitude and a direction. A small compass placed at a point tends to align with the local field direction. A moving positive test charge experiences the magnetic part of the Lorentz force according to

$𝐅_{B} = q 𝐯 \times 𝐁 .$

The magnitude is

$F_{B} = | q | v B \sin θ,$

where $θ$ is the angle between the velocity $𝐯$ and the field $𝐁$ . The force is greatest when the velocity is perpendicular to the field and zero when the velocity is parallel or antiparallel to the field.

This compass-board image shows how many local direction measurements can be combined into a field-line map. A single compass reports only the direction at its own position; a field diagram summarizes many such measurements.

Field-Line Conventions

Field lines are drawn so that the tangent to a line points in the direction of $𝐁$ . Where the lines are denser, the field is represented as stronger. Magnetic field lines never terminate on an isolated magnetic charge in ordinary classical electromagnetism. Instead, they form closed loops, consistent with Gauss's law for magnetism.

A useful reading strategy is to separate three questions: Where is the field strongest? Which way does the vector point? How does the geometry reflect the source? This prevents a common misconception that a field line is the path a charged particle must follow. A particle path depends on the particle's velocity, charge, mass, and the local field.

Direction Symbols and Right-Hand Rules

In two-dimensional diagrams, a dot often represents a vector pointing out of the page, like the tip of an arrow coming toward you. A cross represents a vector pointing into the page, like the tail feathers of an arrow moving away from you.

For a straight current-carrying wire, point your right thumb in the direction of conventional current. Your curled fingers then show the circular direction of the magnetic field around the wire.

For a positive moving charge, orient your right hand according to the cross product $𝐯 \times 𝐁$ . A negative charge experiences force in the opposite direction.

Magnetic Force on Moving Charges

The Lorentz Force

The full electromagnetic force on a charge is

$𝐅 = q (𝐄 + 𝐯 \times 𝐁) .$

The electric component can change the particle's kinetic energy directly. The magnetic component is perpendicular to the instantaneous velocity, so for a point charge it changes the direction of motion rather than the speed.

If a nonrelativistic charged particle enters a uniform magnetic field with $𝐯 ⊥ 𝐁$ , the magnetic force acts as a centripetal force. The path is circular, with radius

$r = \frac{m v}{| q | B} .$

If the velocity has both perpendicular and parallel components, the idealized trajectory becomes a helix. These ideas are central to mass spectrometry, cyclotron motion, and charged-particle beam control.

Problem-Solving Pattern

When you solve a magnetic-force problem, first identify the sign of the charge, then find the angle between velocity and field, calculate the magnitude, and finally determine the direction. For a negative charge, reverse the direction obtained from the right-hand rule for a positive charge.

A useful check is dimensional: since $1 T = 1 N / (A m)$ , the expression $q v B$ has the unit newton.

Currents as Sources of Magnetic Fields

Long Straight Conductors

A steady current in a long straight conductor produces circular magnetic field lines. In vacuum or approximately in air, the magnitude at perpendicular distance $r$ from an ideal long straight wire is

$B = \frac{μ_{0} I}{2 π r} .$

Here $I$ is the current and $μ_{0}$ is the magnetic constant, approximately $4 π \times 1 0^{- 7} T m / A$ . The result shows two important proportionalities: doubling the current doubles the field, while doubling the distance halves the field.

The Biot–Savart Law

For a steady current with arbitrary wire geometry, the Biot–Savart law adds the vector contributions from small current elements:

$𝐁 (𝐫) = \frac{μ_{0}}{4 π} \int \frac{I d ℓ \times \hat{𝐫}}{r^{2}} .$

The law is a vector superposition rule. The current element $d ℓ$ points in the conventional-current direction, while $\hat{𝐫}$ points from the current element toward the observation point. For highly symmetric systems, Ampère's law is often more efficient.

Circular Current Loops

At the center of a circular loop of radius $R$ carrying current $I$ , the field magnitude for one turn is

$B = \frac{μ_{0} I}{2 R} .$

For $N$ closely stacked turns, the field at the common center is approximately $N$ times larger. Curl the fingers of your right hand in the current direction; your thumb gives the field direction along the loop axis.

The field around a current loop has a dipole-like shape. A current loop therefore behaves, at distances large compared with its size, much like a magnetic dipole.

This computed iron-filings visualization emphasizes that field direction and field strength vary throughout space, even for a geometrically simple source.

Solenoids

A solenoid is a long helical coil. Near the center of an ideal long air-core solenoid, the magnetic field is approximately uniform and parallel to the axis. Its magnitude is

$B \approx μ_{0} n I,$

where $n = N / L$ is the number of turns per unit length. Increasing the current or the turn density increases the field.

The same right-hand idea applies to the whole coil: curl your fingers in the direction of conventional current around the turns, and your thumb points along the field inside the solenoid.

Ampère's Law and Symmetry

For steady currents, Ampère's circuital law can be written as

$\oint 𝐁 \cdot d ℓ = μ_{0} I_{e n c} .$

The line integral is taken around a closed path, and $I_{e n c}$ is the current passing through a surface bounded by that path. Ampère's law is especially powerful when symmetry makes the magnitude of $𝐁$ constant along suitable parts of the path, as for an ideal long straight wire, an ideal long solenoid, or a toroid.

In time-dependent situations, the more general Ampère–Maxwell law also contains a term related to changing electric flux. This extension is one of Maxwell's equations and shows that changing electric fields can also act as sources of magnetic fields.

Magnetic Force on Conductors and Current Loops

Force on a Straight Current-Carrying Wire

A straight wire segment of vector length $𝐋$ , carrying current $I$ in a uniform external field, experiences the magnetic force

$𝐅 = I 𝐋 \times 𝐁 .$

Its magnitude is $F = I L B \sin θ$ . The direction follows the same cross-product logic as the force on moving positive charges. This connection is not accidental: electric current is organized motion of charge.

Torque and Magnetic Dipole Moment

A planar current loop with $N$ turns, area $A$ , and current $I$ has magnetic dipole moment

$𝝁 = N I A \hat{𝐧},$

where $\hat{𝐧}$ is normal to the loop according to a right-hand rule. In a uniform magnetic field,

$𝝉 = 𝝁 \times 𝐁$

and the potential energy is

$U = - 𝝁 \cdot 𝐁 .$

This torque principle is fundamental to electric motors and many measuring instruments. A uniform field can give a loop zero net force while still exerting a nonzero torque.

Magnetic Flux and Gauss's Law for Magnetism

Magnetic flux through a surface measures how much of the magnetic field passes through that surface:

$Φ_{B} = \int 𝐁 \cdot d 𝐀 .$

For a uniform field through a flat area $A$ ,

$Φ_{B} = B A \cos θ,$

where $θ$ is the angle between $𝐁$ and the surface normal. The SI unit of magnetic flux is the weber (Wb).

For any closed surface,

$\oint 𝐁 \cdot d 𝐀 = 0.$

This is Gauss's law for magnetism. It expresses the fact that the net magnetic flux through a closed surface is zero. In classical electromagnetism, magnetic field lines form closed loops; isolated magnetic monopoles have not been experimentally established.

Magnetic flux becomes especially important when it changes with time, because changing flux is linked to induced electric fields by Faraday's law of induction.

Magnetic Materials

Matter responds to magnetic fields because electrons carry orbital and spin magnetic moments.

Diamagnetic materials develop a weak induced response opposing the applied field. Paramagnetic materials develop a weak response tending to align with the applied field. Ferromagnetic materials can show strong cooperative alignment of microscopic magnetic moments and can retain magnetization after an external field is removed.

In ferromagnets, regions called magnetic domains can become preferentially aligned. The relation between applied field and magnetization can show hysteresis, which matters in transformer cores, motors, magnetic recording, and permanent magnets.

A material's magnetic response depends on temperature, composition, microstructure, and field history. For advanced work, distinguish the magnetic flux density $𝐁$ from the magnetic field strength $𝐇$ , especially inside matter.

Earth's Magnetic Field

Earth is surrounded by a magnetic field that is approximately dipolar near the planet, although the real field is more complicated and changes with location and time. The field is generated mainly by the geodynamo: motion of electrically conducting liquid metal in Earth's outer core sustains electric currents and magnetic fields.

A compass aligns approximately with the horizontal component of the local geomagnetic field. Geographic north and magnetic north are not identical, and magnetic declination changes with position and time. Earth's surface field is typically on the order of tens of microteslas.

Measuring and Exploring Magnetic Fields

Magnetic fields can be investigated with compasses, Hall sensors, search coils, magnetometers, and calibrated laboratory probes. A compass is mainly a directional indicator; a Hall sensor can provide an electrical signal related to field magnitude and direction.

When collecting data, you should control distance, orientation, current, and sensor zero offset. Record uncertainties and repeat measurements. A field map is stronger evidence when it combines many measured points with a clearly stated coordinate system.

For an interactive model, explore the PhET Faraday's Electromagnetic Lab. Use the bar-magnet and electromagnet screens to predict field direction before turning on field indicators, then compare your prediction with the simulation.

Applications and Connections

Electric motors use magnetic forces and torques on current-carrying conductors. Generators use changing magnetic flux to produce induced electromotive force. Loudspeakers use forces on current-carrying coils. Hall sensors detect magnetic fields through charge separation in a conductor or semiconductor. Particle accelerators use magnetic fields to steer charged beams. Magnetic resonance imaging uses strong magnetic fields together with field gradients and radio-frequency excitation to create medical images.

These examples connect electromagnetism to engineering, medical physics, geophysics, astronomy, materials science, and electronics. The same field concept also prepares you for a deeper study of Maxwell's equations and electromagnetic waves.

Interactive Tasks

Quiz: Test Your Knowledge

Which SI unit is used for the magnetic field B? (Tesla) (!Weber) (!Volt) (!Coulomb)

When is the magnetic force on a moving charged particle largest for fixed speed and field strength? (When velocity is perpendicular to the magnetic field) (!When velocity is parallel to the magnetic field) (!When the particle is at rest) (!When velocity is antiparallel to the magnetic field)

What direction does the magnetic force on a positive charge have relative to velocity and magnetic field? (It is perpendicular to both) (!It is parallel to velocity) (!It is always parallel to the magnetic field) (!It is always opposite to velocity)

How does the magnetic field of an ideal long straight wire vary with distance from the wire? (It is inversely proportional to distance) (!It is directly proportional to distance) (!It is independent of distance) (!It is inversely proportional to distance squared)

What happens to the field near the center of an ideal long solenoid if the current doubles? (The field approximately doubles) (!The field approximately halves) (!The field becomes zero) (!The field becomes four times larger)

Which quantity has the SI unit weber? (Magnetic flux) (!Magnetic field) (!Electric current) (!Magnetic force)

What does Gauss's law for magnetism state about a closed surface? (The net magnetic flux through it is zero) (!The magnetic field is zero everywhere on it) (!The electric current through it must be zero) (!The magnetic flux is always positive)

What is the path of a nonrelativistic charged particle moving perpendicular to a uniform magnetic field if no other force acts? (A circle) (!A straight line) (!A parabola) (!A stationary point)

Which expression gives the magnetic force on a straight current-carrying wire in a uniform field? (Current times vector length crossed with magnetic field) (!Charge divided by magnetic field) (!Voltage times resistance) (!Magnetic flux divided by area only)

What can a uniform magnetic field exert on a closed current loop even when the net force is zero? (A torque) (!A net electric charge) (!A gravitational field) (!A permanent increase in current)

Memory Game

Tesla	SI unit of magnetic flux density
Lorentz force	Force on a charged particle due to electric and magnetic fields
Biot-Savart law	Rule for adding field contributions from steady current elements
Ampere's law	Closed-path relation between magnetic circulation and enclosed steady current
Solenoid	Helical coil that can produce an approximately uniform internal field
Magnetic flux	Surface integral of the normal component of a magnetic field
Dipole moment	Vector that determines the torque of a current loop in an external field

Drag and Drop

Match the correct terms.	Topic
Circular field around the conductor	Long straight current-carrying wire
Approximately uniform internal field	Long ideal solenoid
Dipole-like axial field	Circular current loop
Perpendicular magnetic force	Moving charged particle
Zero net flux through a closed surface	Gauss's law for magnetism

...

Crossword Puzzle

Tesla	What is the SI unit of magnetic field B?
Lorentz	Which surname is associated with the force law for charges in electric and magnetic fields?
Solenoid	What coil can produce an approximately uniform magnetic field inside?
Ampere	Which scientist's name is used for the circuital law relating magnetic field to current?
Dipole	What model describes a small current loop or bar magnet at large distances?
Flux	What four-letter word describes the surface integral of a field through an area?

LearningApps

Cloze Text

A magnetic field is represented by the vector

. Its SI unit is the

. A moving charge experiences a magnetic force proportional to the sine of the angle between its velocity and the

. For a positive charge, the force direction follows the cross product of velocity with the

. A long straight current produces circular field lines whose magnitude decreases with

. Near the center of an ideal long solenoid, the field is approximately proportional to current and turn

. Magnetic flux through a flat surface depends on the component of the field along the surface

. Gauss's law for magnetism states that the net magnetic flux through a closed surface is

. A current loop has a magnetic dipole moment and can experience a

in a uniform field. Earth's large-scale field is sustained mainly by the conducting fluid motion of the outer-core

.

Open-Ended Tasks

Easy

Magnetic field mapping: Use a bar magnet and a small compass to draw a field-direction map on paper; photograph the final map and explain how your measured directions relate to field lines.
Magnetic field vocabulary: Create a one-page illustrated explainer that connects field, tesla, flux, force, current, and dipole using your own examples.
Right-hand rule: Produce a short photo sequence or video that demonstrates the right-hand rule for a straight current and the rule for a positive moving charge.
Physics simulation: Use the PhET Faraday's Electromagnetic Lab to make three predictions about field direction or strength, test them, and write a short comparison of prediction and observation.

Standard

Current and magnetic field experiment: With teacher-approved low-voltage equipment, investigate how a compass or field sensor reading changes when current in a straight conductor changes; graph the result and discuss uncertainty.
Solenoid investigation: Build or use a classroom solenoid and compare field strength for different currents or turn densities while keeping other variables controlled.
Magnetic force video analysis: Record a safe classroom demonstration of a current-carrying wire or coil in an external field, annotate the force direction frame by frame, and explain it with vector reasoning.
Magnetism interview: Interview an engineer, technician, teacher, medical physicist, or other relevant professional about one practical use of magnetic fields and turn the interview into a two-minute audio or video report.

Advanced

Magnetometer project: Use a calibrated sensor or smartphone magnetometer to map how a magnet's measured field varies with position, model the trend, and evaluate limitations of your measurement method.
Biot-Savart model: Create a spreadsheet or program that numerically adds contributions from current elements in a loop and compare the calculated axial field with an analytical result at a selected point.
Ampere's law reasoning: Design a poster or digital presentation that explains why symmetry makes Ampere's law efficient for a long wire or ideal solenoid and why the same method is difficult for an arbitrary current shape.
Physics field study: Visit a science museum, university laboratory, engineering workshop, or other relevant learning site and produce a report linking at least three observed technologies to magnetic-field principles.

Learning Assessment

Vector reasoning assessment: Analyze a charged particle entering a uniform magnetic field at an oblique angle; predict the three-dimensional path and justify how the speed and direction change.
Model comparison assessment: Compare the magnetic fields of a bar magnet, current loop, and solenoid; identify similarities, important differences, and the limits of the dipole analogy.
Experimental design assessment: Design a method to test the inverse-distance relation for the field around a long straight conductor, including controls, safety, uncertainty, and a plan for data analysis.
Application assessment: Explain how the same magnetic force principles appear in both a loudspeaker and an electric motor, while identifying what differs in their design goals.
Flux transfer assessment: Given a coil whose area and orientation can change in a magnetic field, reason qualitatively about how each change alters magnetic flux and what this implies for later study of induction.
Evidence evaluation assessment: Evaluate whether an iron-filings image alone is enough to determine the numerical strength of a magnetic field; propose additional measurements needed for a quantitative conclusion.

Evidence of Learning

Knowledge: You can explain magnetic fields as vector fields, distinguish field from force and flux, and connect currents, moving charges, magnetic dipoles, materials, and Earth's geodynamo to magnetic phenomena.

Skills: You can use right-hand rules, vector products, field-line diagrams, magnetic-field equations, proportional reasoning, graphing, uncertainty analysis, and appropriate digital or laboratory measurement tools.

Products: Strong evidence may include a measured field map, an annotated diagram, a simulation report, a graph from an experiment, a numerical model, a presentation, or a short explanatory video.

Transfer achievements: You can use the same principles to reason about unfamiliar technologies such as motors, sensors, speakers, particle-beam systems, and magnetic imaging, while identifying which assumptions of an idealized model are valid.