Math 2.0

Optics, electromagnetic theory Definition

Magnetic field

Builds on Electric chargeElectric fieldVector fieldCross productMaxwell's equations

Definition

Let U⊆ℝ3U \subseteq \R^3 be a region of space and I⊆ℝI \subseteq \R an interval of time. The magnetic field in UU is a vector field B ⁣:U×I→ℝ3B \colon U \times I \to \R^3 with the following property. Let a point charge qq be at x∈Ux \in U at time t∈It \in I, moving with velocity v∈ℝ3v \in \R^3. Besides the electric force qE(x,t)q E(x, t), it feels the magnetic force Fmag=q v×B(x,t),F_{\mathrm{mag}} = q \, v \times B(x, t), where ×\times is the cross product and EE is the electric field.

Remarks

The field BB is determined by this property: the forces for velocities v=e1v = e_1 and v=e2v = e_2 give the components of BB orthogonal to e1e_1 and to e2e_2, hence all of BB. Together with EE, the field BB satisfies Maxwell's equations. Example: the field B(x)=(0,0,B0)B(x) = (0, 0, B_0) is a uniform magnetic field of strength B0B_0 along the third axis; a charge moving along the first axis is pushed along the second axis. Non-example: the vector field B(x)=xB(x) = x is not a possible magnetic field, since its divergence is 3≠03 \ne 0, while every magnetic field is divergence-free. Some texts call BB the magnetic flux density and reserve "magnetic field" for H=B/μ0−MH = B / \mu_0 - M, where MM is the magnetization. The SI unit of BB is the tesla.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptElectric charge, concept, no evidence levelElectric chargeconceptElectric field, concept, no evidence levelElectric fieldconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptMaxwell's equations, concept, no evidence levelMaxwell's equationsconceptVector-valued function, concept, no evidence levelVector-valued functionconceptVector field, concept, no evidence levelVector fieldconceptMagnetic field, definition, no evidence levelMagnetic fielddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cross product concept
  2. Electric charge concept
  3. Electric field concept
  4. Euclidean space object
  5. Function concept
  6. Maxwell's equations concept
  7. Vector-valued function concept
  8. Vector field concept
  9. Magnetic field definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Magnetic_field
About this page
TagTZYZ78HRY2
URIm2://TZYZ78HRY2
Kinddefinition
WritingW0 stub
MSC78A30, 78A25
DefinesMagnetic field
UsesElectric charge, Electric field, Vector field, Cross product, Maxwell's equations
Changed3 November 2026

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