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PhysicsDifficulty 2-4

Magnetism

Magnetism is the push and pull between magnets, and between moving charges, carried by a magnetic field measured in teslas. Every magnet has a north and a south pole that cannot be separated; like poles repel and unlike poles attract. Electric currents create magnetic fields, and changing magnetic fields create currents — the link that powers generators and, together with electricity, explains light.

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Sources of Magnetic Field

Integrating the Biot–Savart law along an infinite straight wire gives a field that circles the wire and falls off as 1/R. Applying Ampère's law to a long solenoid gives a uniform interior field proportional to the turns per unit length.

Bwire=μ0I2πR,Bsolenoid=μ0nIB_{\text{wire}} = \dfrac{\mu_0 I}{2\pi R}, \qquad B_{\text{solenoid}} = \mu_0 n I

Long straight wire (left) and the interior of a long solenoid (right).

With μ₀ = 1.25663706127 × 10⁻⁶ N A⁻² (CODATA 2022), 10 A at 5.0 cm gives B ≈ 4.0 × 10⁻⁵ T. The vector force on a moving charge is F = qv × B; because F ⟂ v, F·v = 0 and the field does no work.

ε=−NdΦmdt,Φm=∫SB⋅n^ dA\varepsilon = -N\dfrac{d\Phi_m}{dt}, \qquad \Phi_m = \int_S \mathbf{B}\cdot\hat{\mathbf{n}}\,dA

Faraday's law with magnetic flux defined as the surface integral of B.

Info: Lenz's law is energy conservation in disguise: pushing a magnet into a coil, you work against the field of the induced current, and that work becomes the current's electrical energy.
Full explanation — the complete reference version every reading depth is based on

Magnets and poles

  • Every magnet has two poles, north (N) and south (S). Like poles repel; unlike poles attract; either pole attracts unmagnetised iron.
  • You cannot get a lone pole: cut a bar magnet in half and each half is a complete magnet with its own N and S.
  • Magnetic field lines show the field's direction (from N to S outside a magnet) and strength (closer lines, stronger field). They never cross and always form closed loops.
  • Earth acts like a giant bar magnet. The magnetic pole near the geographic North Pole is in fact a south magnetic pole — which is exactly why the north end of a compass needle swings towards it.

Currents make magnetic fields

In 1820 Hans Christian Oersted reported that a compass needle moved whenever current flowed in a nearby wire: an electric current produces a magnetic effect, and moving charges create magnetic fields. Around a long straight wire the field lines are circles centred on the wire, and the field weakens with distance.

B=μ0I2πRB = \dfrac{\mu_0 I}{2\pi R}

Magnetic field a distance R from a long straight wire carrying current I.

Worked example: a wire and a compass

Worked example (our calculation): with μ₀ ≈ 1.2566 × 10⁻⁶ N A⁻², a wire carrying 10 A produces B = (1.2566 × 10⁻⁶ × 10)/(2π × 0.05) ≈ 4.0 × 10⁻⁵ T = 40 μT at 5 cm. That is comparable to Earth's own field (about 22–67 μT at the surface), which is why a nearby current visibly turns a compass. Winding wire into a coil (a solenoid) concentrates the field: inside a long solenoid B = μ₀nI, uniform across the interior.

Forces on moving charges

F=qvBsin⁡θF = qvB\sin\theta

Magnetic force on a charge q moving at speed v at angle θ to a field B. 1 T = 1 N/(A·m).

The force is always at right angles to the charge's velocity, so it bends the path without speeding the charge up or slowing it down — the magnetic force does no work. A charge moving straight along the field lines (θ = 0) feels no force at all.

Changing fields make currents

From 1831 Michael Faraday showed the reverse effect: a changing magnetic field through a coil induces an emf. The faster the magnetic flux Φ changes, and the more turns N, the larger the emf. Lenz's law gives the direction: the induced current always opposes the change that caused it, as energy conservation demands. Generators use exactly this, rotating coils in magnetic fields.

ε=−NdΦdt\varepsilon = -N\dfrac{d\Phi}{dt}

Faraday's law of induction; the minus sign expresses Lenz's law.

Magnetic materials and Earth's field

Iron and its alloys are ferromagnetic: they contain tiny regions called domains in which atomic magnets are lined up. In an unmagnetised piece the domains point every which way; a strong field can line them up. Earth's own field is generated in its fluid outer core by a self-sustaining dynamo of currents in moving molten iron, and rocks record that its polarity has reversed many times, slowly and irregularly.

  • Earth's field at the surface: about 22,000–67,000 nT (22–67 μT), often rounded to 5 × 10⁻⁵ T.
  • The strongest permanent magnets: near 2 T.
  • Superconducting electromagnets: 10 T or more.

Where magnetism connects

Magnetism builds on Electricity (currents are its source) and Forces (F = qvB sin θ is a force like any other). Gravity, by contrast, is always attractive (see Gravity). Maxwell's unification of electricity and magnetism predicted electromagnetic waves travelling at c = 1/√(ε₀μ₀) — the explanation of Light.

How we know

  • Oersted's 1820 report of a compass needle moving whenever current flowed in a nearby wire showed that an electric current has a magnetic effect.
  • Faraday's experiments with changing magnetic fields, from 1831, established electromagnetic induction.
  • Geomagnetic observatories record accurate measurements of Earth's field direction and strength, continuously, over many years.
  • The magnetisation locked into rocks from many sites around the world records the direction of Earth's past field — the evidence for its reversals.
Common misconception: Misconception: Earth's 'North Magnetic Pole' must be a north magnetic pole. Opposites attract, so the pole that pulls the north end of a compass towards it must be a south magnetic pole. The name describes where it is, not its polarity.
Common misconception: Misconception: magnets attract all metals. Magnets strongly attract ferromagnetic materials such as iron. Aluminium (paramagnetic) and copper (diamagnetic) respond so weakly — their magnetic susceptibilities are only about 2.2 × 10⁻⁵ and −9.7 × 10⁻⁶ — that a fridge magnet will not stick to them.
Info: Assumptions and limits: B = μ₀I/(2πR) is exact only for an infinitely long, thin, straight wire, and B = μ₀nI only for the middle of a long solenoid, away from its ends. Earth's field is not a perfect bar magnet's either: its strength varies from place to place (about 22–67 μT) and it changes slowly over time. Strongly magnetic behaviour is a property of ferromagnetic materials such as iron; most other materials respond only very weakly.

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Content status: published 1 October 2026.

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