6.1 Magnetic fields

Magnetic fields describe the forces that act on moving charges and magnetic materials. This topic connects electric currents to the magnetic forces they produce, building toward a full picture of electromagnetism. You'll see these ideas show up everywhere from electric motors to MRI machines, so getting comfortable with them now pays off throughout the rest of Physics II.

Fundamental concepts of magnetism

Magnetism arises from the motion of electric charges. Any time a charge moves, it creates a magnetic field in the surrounding space. This connection between electricity and magnetism is the foundation of electromagnetism.

Magnetic field definition

A magnetic field is a vector field that describes the region of space where magnetic forces act on moving charged particles or magnetic materials. It's denoted by B and measured in tesla (T).

Magnetic fields originate from two sources: moving electric charges (currents) and the intrinsic magnetic moments of elementary particles like electrons. You can visualize the field using magnetic field lines, which show both the direction and relative strength of the field at each point.

Magnetic field lines

Magnetic field lines are an imaginary tool for representing the direction and strength of a magnetic field. A few key rules govern them:

• They always form closed loops (they never start or end in empty space)
• They never intersect each other
• Where lines are packed more closely together, the field is stronger
• The field vector at any point is tangent to the field line passing through that point
• By convention, field lines emerge from the north pole and enter the south pole of a magnet

These rules help you predict how magnetic materials and charged particles will behave in a given field.

Magnetic flux

Magnetic flux ($\Phi_B$) measures how much magnetic field passes through a given surface. Think of it as counting how many field lines thread through an area.

$\Phi_B = \int \mathbf{B} \cdot d\mathbf{A}$

For a uniform field passing straight through a flat surface, this simplifies to $\Phi_B = BA\cos\theta$, where $\theta$ is the angle between the field and the surface normal.

• Measured in webers (Wb), where 1 Wb = 1 T·m²
• Magnetic flux is central to electromagnetic induction and shows up directly in Faraday's law

Sources of magnetic fields

Magnetic fields can come from permanent magnets, electric currents, or natural sources like the Earth's core. Each source produces fields with distinct characteristics.

Permanent magnets

Permanent magnets produce their own persistent magnetic field without any external power source. They're made of ferromagnetic materials (like iron, nickel, or cobalt) whose internal magnetic domains are aligned in the same direction.

• Every permanent magnet has a north pole and a south pole
• Field lines emerge from the north pole and enter the south pole
• Their strength depends on the material composition and how they were manufactured
• They can be weakened or demagnetized by heat, strong external fields, or physical shock

Electric currents

Moving electric charges generate magnetic fields perpendicular to their direction of motion. This is the principle behind electromagnets: by running current through a wire, you create a controllable magnetic field.

• The field strength depends on the current magnitude and the distance from the conductor
• The right-hand rule determines the field direction: wrap your right hand around the wire with your thumb pointing in the current direction, and your fingers curl in the direction of the field
• Unlike permanent magnets, electromagnets can be turned on and off and adjusted in strength, which is why they're used in MRI machines, particle accelerators, and countless other devices

Earth's magnetic field

Earth's magnetic field (the geomagnetic field) is generated by electric currents flowing in the planet's liquid outer core. It has a roughly dipole structure, with magnetic poles near (but not exactly at) the geographic poles.

• Field strength varies across the globe, typically ranging from about 25 to 65 microteslas
• It shields the planet from harmful solar radiation and charged particles
• The magnetic poles undergo periodic reversals over geological timescales
• Many animal species and human navigation technologies rely on this field

Magnetic force

Magnetic forces act on moving charged particles and current-carrying conductors. These forces are always perpendicular to the velocity of the charge, which means magnetic fields can change a particle's direction but never its speed.

Force on moving charges

A charged particle moving through a magnetic field experiences a force given by:

$\mathbf{F} = q\mathbf{v} \times \mathbf{B}$

The magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between the velocity and the field.

Key features of this force:

• It's perpendicular to both the velocity and the magnetic field
• It does no work on the particle (since it's always perpendicular to motion), so kinetic energy stays constant
• Direction is found using the right-hand rule for positive charges (point fingers along v, curl toward B, and your thumb gives the force direction). For negative charges, the force points the opposite way.
• In a uniform field, this perpendicular force causes charged particles to follow circular or helical paths

This principle is used in mass spectrometers to separate ions by their mass-to-charge ratio.

Force on current-carrying wires

A straight current-carrying wire in a magnetic field experiences a force:

$\mathbf{F} = I\mathbf{L} \times \mathbf{B}$

The magnitude is $F = ILB\sin\theta$, where $I$ is the current, $L$ is the wire length, and $\theta$ is the angle between the current direction and the field.

• The force direction follows the right-hand rule (point fingers along the current, curl toward B)
• This is the operating principle behind electric motors and loudspeakers, which convert electrical energy into mechanical motion

Lorentz force equation

When both electric and magnetic fields are present, the total force on a charged particle is the Lorentz force:

$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$

This combines the electric force ($q\mathbf{E}$) and the magnetic force ($q\mathbf{v} \times \mathbf{B}$) into a single expression. It's the fundamental equation governing charged particle motion in electromagnetic environments, from particle accelerators to fusion reactors.

Magnetic field strength

Magnetic field strength quantifies how intense a magnetic field is at a given point. Getting the units and measurement methods straight is important for comparing fields across very different scales.

Units of magnetic field

• Tesla (T) is the SI unit, defined as 1 T = 1 N/(A·m)
• Gauss (G) is the CGS unit, with 1 T = 10,000 G
• Earth's magnetic field is on the order of 25–65 microteslas (μT)
• An MRI machine typically operates at 1.5 to 3 T
• Research magnets can reach 45 T or more
• Nanotesla (nT) is commonly used for weak fields in space physics and geophysics

Measuring magnetic fields

Different instruments are used depending on the field strength and required precision:

• Hall effect sensors detect the voltage induced across a conductor placed in a magnetic field. These are common and relatively inexpensive.
• Fluxgate magnetometers measure fields by detecting changes in the magnetic saturation of a core material. Used in geophysics and spacecraft.
• SQUIDs (Superconducting Quantum Interference Devices) offer extremely sensitive detection, capable of measuring fields as weak as a few femtoteslas.
• Compass needles provide a simple way to detect field direction in weak fields.
• NMR (Nuclear Magnetic Resonance) enables precise measurement of strong, uniform fields.

Magnetic field intensity

The magnetic field intensity $\mathbf{H}$ represents the strength of a magnetic field independent of the material it's passing through. It's measured in amperes per meter (A/m).

$\mathbf{H}$ relates to the magnetic flux density $\mathbf{B}$ through:

$\mathbf{B} = \mu \mathbf{H}$

where $\mu$ is the permeability of the material. This distinction between $\mathbf{B}$ and $\mathbf{H}$ becomes important when you analyze magnetic fields inside different materials, because the same $\mathbf{H}$ can produce very different $\mathbf{B}$ values depending on the medium.

Magnetic materials

Different materials respond to magnetic fields in very different ways, depending on their atomic and electronic structure. These responses fall into three main categories.

Ferromagnetism

Ferromagnetic materials (iron, nickel, cobalt) exhibit strong magnetic properties and can maintain magnetization even after an external field is removed. This happens because unpaired electron spins in these materials spontaneously align parallel to each other within regions called magnetic domains.

• Each domain is a microscopic region where all the magnetic moments point the same direction
• When an external field aligns many domains, the material becomes strongly magnetized
• Ferromagnets display hysteresis: their magnetization depends on their magnetic history, not just the current applied field
• This property enables permanent magnets and magnetic data storage (hard drives)

Paramagnetism vs diamagnetism

Paramagnetic materials (aluminum, platinum) have unpaired electrons that weakly align with an applied field, causing a slight attraction. Diamagnetic materials (copper, water, bismuth) have no unpaired electrons; instead, the applied field induces tiny opposing magnetic moments, causing a slight repulsion.

• Both effects are much weaker than ferromagnetism
• Paramagnetic susceptibility is small and positive; diamagnetic susceptibility is small and negative
• Diamagnetism is actually present in all materials, but it's usually overshadowed by stronger paramagnetic or ferromagnetic effects when those are present

Magnetic domains

Within a ferromagnetic material, magnetic domains are microscopic regions where all the atomic magnetic moments are aligned in the same direction. Domains form because this arrangement minimizes the material's overall magnetic energy.

• In an unmagnetized piece of iron, domains point in random directions, so their fields cancel out
• Applying an external field causes favorable domains to grow (through domain wall motion) and unfavorable ones to shrink
• At high enough fields, all domains align and the material reaches magnetic saturation
• Understanding domain behavior is key to designing better magnetic materials for specific applications

Electromagnetic induction

Electromagnetic induction is the process of generating an electric current by changing the magnetic flux through a conductor. This is how most of the world's electricity is produced.

Faraday's law

Faraday's law states that the induced electromotive force (emf) in a closed loop equals the negative rate of change of magnetic flux through the loop:

$\varepsilon = -\frac{d\Phi_B}{dt}$

The negative sign reflects Lenz's law (see below). Faraday's law applies whether the flux changes because the conductor moves, the magnetic field changes, or both.

• This is the principle behind generators (mechanical energy → electrical energy)
• It also underlies induction cooking, wireless charging, and transformers

Lenz's law

Lenz's law states that the direction of an induced current always opposes the change in magnetic flux that produced it. If the flux through a loop is increasing, the induced current creates a field that opposes the increase. If the flux is decreasing, the induced current tries to maintain it.

• This is a direct consequence of conservation of energy: if the induced current reinforced the change, you'd get energy from nothing
• Lenz's law explains the braking effect in magnetic damping systems (like eddy current brakes)
• It's essential for predicting current direction in inductors and transformers

Motional emf

When a conductor moves through a magnetic field, the free charges inside it experience a magnetic force that drives them along the conductor, creating a motional emf:

$\varepsilon = vBL$

for a straight conductor of length $L$ moving at velocity $v$ perpendicular to a uniform field $B$. More generally:

$\varepsilon = -\int (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l}$

This demonstrates the direct conversion between mechanical and electrical energy and is the working principle behind generators, electromagnetic flow meters, and certain types of sensors.

Applications of magnetic fields

Magnetic fields are at the heart of many technologies you encounter daily. The concepts from this unit connect directly to how these devices work.

Electric motors

Electric motors convert electrical energy into mechanical energy using the force on current-carrying conductors in a magnetic field.

• A motor consists of a rotor (the rotating part) and a stator (the stationary part), with electromagnets or permanent magnets providing the field
• Current through the rotor coils creates a torque that drives rotation
• Common types include DC motors, AC induction motors, and brushless DC motors
• Found in electric vehicles, industrial machinery, household appliances, and countless other products

Generators

Generators do the reverse of motors: they convert mechanical energy into electrical energy through electromagnetic induction.

• A coil rotates in a magnetic field (or a magnetic field rotates around a coil), and the changing flux induces an emf
• They can produce AC or DC depending on the design
• Nearly all large-scale electricity production uses generators (hydroelectric, thermal, wind)
• Vehicle alternators are small generators that charge the battery and power electrical systems

Magnetic levitation

Magnetic levitation uses magnetic fields to suspend objects without physical contact, eliminating friction.

• Can work through repulsion between like magnetic poles or through induced eddy currents in a conductor
• Maglev trains use this principle for high-speed, low-friction transportation (some reach over 600 km/h)
• Magnetic bearings in turbines and industrial equipment reduce wear and energy loss

Magnetic field interactions

Magnetic fields from different sources combine and interact, and understanding these interactions is essential for analyzing real electromagnetic systems.

Magnetic field superposition

The principle of superposition applies to magnetic fields: when multiple sources are present, the total field at any point is the vector sum of the individual fields.

$\mathbf{B}_{total} = \mathbf{B}_1 + \mathbf{B}_2 + \mathbf{B}_3 + \cdots$

This allows you to analyze complex configurations by breaking them into simpler pieces. It's used in designing electromagnets with multiple coils and in magnetic shielding, where fields are arranged to cancel each other out.

Magnetic dipoles

A magnetic dipole is the simplest magnetic field source, consisting of a north and south pole separated by a small distance. It's characterized by its magnetic dipole moment $\mathbf{m}$, a vector pointing from the south pole to the north pole.

• Bar magnets, compass needles, and current loops all behave as magnetic dipoles
• In an external field, a dipole experiences both a force (if the field is non-uniform) and a torque (which tends to align it with the field)
• The dipole model is also the starting point for understanding the magnetic properties of individual atoms and molecules

Torque on current loops

A current-carrying loop placed in a magnetic field experiences a torque that tends to align the loop's area vector with the field:

$\tau = NIAB\sin\theta$

where $N$ is the number of turns, $I$ is the current, $A$ is the loop area, $B$ is the field strength, and $\theta$ is the angle between the loop's area vector and the field.

• Maximum torque occurs when $\theta = 90°$ (loop perpendicular to the field)
• Zero torque when $\theta = 0°$ (loop aligned with the field)
• This is the operating principle of galvanometers (sensitive current meters) and electric motors

Ampère's law

Ampère's law relates the magnetic field along a closed loop to the total current passing through that loop:

$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$

It's most useful for calculating magnetic fields in situations with high symmetry (straight wires, solenoids, toroids).

Magnetic field of a straight wire

For a long, straight wire carrying current $I$, the magnetic field forms concentric circles around the wire. Applying Ampère's law with a circular path of radius $r$ gives:

$B = \frac{\mu_0 I}{2\pi r}$

where $\mu_0 = 4\pi \times 10^{-7}$ T·m/A is the permeability of free space. The field strength decreases as $1/r$, so it drops off with distance from the wire. Use the right-hand rule to find the field direction: thumb along the current, fingers curl in the direction of B.

Magnetic field of a solenoid

A solenoid is a long coil of wire. Inside an ideal solenoid, the magnetic field is nearly uniform and given by:

$B = \mu_0 n I$

where $n$ is the number of turns per unit length and $I$ is the current. Outside the solenoid, the field is approximately zero (for an infinitely long solenoid, it's exactly zero).

• Solenoids are used to generate strong, uniform fields in MRI machines, particle accelerators, and electromagnets
• They also form the basis for inductors in electronic circuits

Magnetic field of a toroid

A toroid is a solenoid bent into a doughnut shape. The magnetic field is confined almost entirely within the toroid and is given by:

$B = \frac{\mu_0 N I}{2\pi r}$

where $N$ is the total number of turns and $r$ is the distance from the center of the toroid. Because the field is contained inside, toroids are useful when you need minimal stray fields. They're commonly found in high-frequency transformers and inductors.

Biot-Savart law

The Biot-Savart law gives the magnetic field produced by an infinitesimal current element at any point in space:

$d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}$

While Ampère's law works best for symmetric situations, the Biot-Savart law can handle any current distribution. You integrate the contributions from every small piece of the conductor to find the total field.

Magnetic field of circular current

For a circular loop of radius $R$ carrying current $I$, the Biot-Savart law gives the field at the center of the loop:

$B = \frac{\mu_0 I}{2R}$

Along the axis of the loop (at distance $x$ from the center):

$B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$

The field is strongest at the center and falls off with distance. This result is the basis for Helmholtz coils, which use two identical loops spaced one radius apart to create a region of nearly uniform field between them.

Magnetic field of arbitrary current distributions

For conductors with irregular shapes, the Biot-Savart law is your go-to tool. The process involves:

1. Break the conductor into infinitesimal elements $d\mathbf{l}$
2. For each element, calculate $d\mathbf{B}$ using the Biot-Savart formula
3. Integrate over the entire conductor to get the total field

This approach is more computationally intensive than Ampère's law but works for any geometry. It's used in designing specialized electromagnets for research and industrial applications.

Comparison with Coulomb's law

The Biot-Savart law for magnetostatics is analogous to Coulomb's law for electrostatics. Both describe how a source (current element or charge) creates a field that falls off as $1/r^2$.

The key difference: the Biot-Savart law involves a cross product ($d\mathbf{l} \times \hat{\mathbf{r}}$), which reflects the fact that magnetic fields wrap around their sources rather than pointing radially outward. This parallel between the two laws hints at the deeper unity of electricity and magnetism captured by Maxwell's equations.

Magnetism in matter

When you place a material in an external magnetic field, the material's atomic structure determines how it responds. The response is characterized by a few key quantities.

Magnetic susceptibility

Magnetic susceptibility ($\chi_m$) measures how strongly a material becomes magnetized in response to an applied field:

$\chi_m = \frac{M}{H}$

where $M$ is the magnetization (magnetic moment per unit volume) and $H$ is the applied field intensity.

• Paramagnetic materials: $\chi_m$ is small and positive (e.g., aluminum, $\chi_m \approx 2 \times 10^{-5}$)
• Diamagnetic materials: $\chi_m$ is small and negative (e.g., copper, $\chi_m \approx -1 \times 10^{-5}$)
• Ferromagnetic materials: $\chi_m$ is large and positive (can be on the order of thousands)
• Superconductors exhibit perfect diamagnetism with $\chi_m = -1$, completely expelling magnetic fields (the Meissner effect)

Permeability of materials

Permeability ($\mu$) describes how easily a material supports the formation of a magnetic field within it:

$\mu = \frac{B}{H}$

It relates to susceptibility through:

$\mu = \mu_0(1 + \chi_m)$

where $\mu_0 = 4\pi \times 10^{-7}$ T·m/A is the permeability of free space.

• High-permeability materials like mu-metal are used for magnetic shielding
• Ferrite cores in transformers and inductors concentrate the magnetic flux, increasing device effectiveness

Hysteresis in ferromagnets

When you magnetize a ferromagnetic material and then reduce the applied field, the magnetization doesn't retrace its original path. This lag is called hysteresis, and plotting magnetization vs. applied field produces a characteristic hysteresis loop.

• The area enclosed by the loop represents energy lost per cycle as heat
• Soft magnetic materials (like silicon steel) have narrow hysteresis loops, meaning low energy loss. These are ideal for transformer cores that cycle many times per second.
• Hard magnetic materials (like neodymium alloys) have wide hysteresis loops, meaning they retain strong magnetization. These are used for permanent magnets and magnetic data storage.