The Magnetic Force and the Biot–Savart Law
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You should be comfortable with vector cross products, line integrals, and the concept of a steady current density \(\vec J\). Familiarity with the operational definition of the magnetic field via the force on a current-carrying wire is assumed.
Why this matters
The Biot–Savart law is the fundamental building block for calculating magnetic fields from arbitrary current distributions.
While electrostatics allows us to calculate fields from charges using Gauss's law, magnetostatics lacks a simple analogue for general geometries because magnetic field lines form closed loops. Instead, we rely on the Biot–Savart law, which expresses the magnetic field generated by a steady current element. This law is derived from the empirical force law between current elements and provides a direct, constructive method to determine \(\vec B\) at any point in space, forming the basis for understanding solenoids, toroids, and the magnetic dipole moment.
The Physical Picture
Imagine a small segment of a wire carrying a steady current. Unlike a point charge, which creates an electric field that radiates outward in all directions, a current element creates a magnetic field that circulates around the wire. The strength of this field depends on how much current is flowing, the length of the segment, and the distance from the segment. Crucially, the direction of the field is not radial; it is perpendicular to both the direction of the current and the line connecting the current element to the point where you are measuring the field. This transverse nature is a defining characteristic of magnetic fields produced by steady currents, distinguishing them fundamentally from electrostatic fields.
From Force to Field
The derivation of the Biot–Savart law begins with the empirical observation of the force between two steady current elements. By defining the magnetic field \(\vec B\) operationally through the force experienced by a test current element, \(d\vec F = I\,d\vec l \times \vec B\), we can isolate the contribution of a source current element to the field at a distant point. The result is an inverse-square law, similar in structure to Coulomb's law, but with a crucial vector cross product that dictates the direction.
Here, \(\vec s = \vec r - \vec r'\) is the vector from the source element \(d\vec l'\) to the field point \(\vec r\), and \(\hat s\) is the corresponding unit vector. The constant \(\mu_0\) is the permeability of free space. The cross product \(d\vec l' \times \hat s\) ensures that the field is zero along the axis of the current and maximum in the plane perpendicular to it.
Geometric Interpretation
The term \(d\vec l' \times \hat s\) can be understood geometrically. The magnitude of the cross product is \(|d\vec l'| \sin\theta\), where \(\theta\) is the angle between the current element and the line of sight to the field point. This means that a current element contributes most strongly to the field at points located in the plane perpendicular to the wire. If you look directly along the wire, the field contribution from that specific element vanishes. The direction of \(d\vec B\) is given by the right-hand rule: if you point your fingers in the direction of the current \(d\vec l'\) and curl them towards \(\hat s\), your thumb points in the direction of \(d\vec B\). This results in field lines that form concentric circles around straight wire segments.
Superposition and Integration
For a continuous current distribution, the total magnetic field is the vector sum of the contributions from all infinitesimal current elements. This is expressed as a line integral over the path of the current. Because the direction of \(d\vec B\) varies from element to element, simple scalar integration is insufficient; one must perform vector integration, often decomposing the field into components. For symmetric geometries, such as a long straight wire or a circular loop, symmetry arguments allow us to eliminate components that cancel out, simplifying the calculation significantly. The Biot–Savart law thus provides a universal tool, though its application requires careful attention to vector geometry.
The derivation
Biot–Savart Law from the Current Force Law
Taking as empirical input the force between two steady current elements (the Grassmann form of the current force law) together with the operational definition of the magnetic field through \(d\vec F = I\,d\vec l \times \vec B\), we derive that a steady current element \(I\,d\vec l'\) at \(\vec r'\) contributes at the field point \(\vec r\) a magnetic field \(d\vec B = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l' \times \hat s}{s^2}\), with \(\vec s = \vec r - \vec r'\), \(s=|\vec s|\), and \(\hat s = \vec s/s\): an inverse-square field, transverse to both the current direction and the line of separation. Integrating over a complete circuit gives the Biot–Savart law.
The Biot–Savart law describes steady currents, and a steady current must flow round a closed circuit. You may split a circuit into segments and add their contributions (that is how the formula is used), but the field of an isolated open segment on its own is not a physical steady-state field: for example, Ampère’s law would not hold for it. Only the sum over the complete circuit is guaranteed to satisfy \(\nabla\cdot\vec B = 0\) and \(\nabla\times\vec B = \mu_0\vec J\). C
Worked example
Problem. Calculate the magnetic field at the center of a circular loop of radius \(R = 0.10\ \mathrm{m}\) carrying a steady current \(I = 5.0\ \mathrm{A}\). Use \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}\).
- Identify the geometry: A circular loop of radius \(R\). The field point is at the center.
- For any current element \(d\vec l'\) on the loop, the vector \(\vec s\) points from the element to the center. The angle between \(d\vec l'\) and \(\vec s\) is always \(90^\circ\), so \(\sin\theta = 1\).
- The magnitude of the contribution from each element is \(dB = \frac{\mu_0}{4\pi} \frac{I\, dl'}{R^2}\). Since the direction of \(d\vec B\) is the same for all elements (perpendicular to the plane of the loop, by the right-hand rule), we can integrate the magnitudes directly.
- Integrate over the circumference: \(B = \int dB = \frac{\mu_0 I}{4\pi R^2} \int dl'\). The integral \(\int dl'\) is the circumference \(2\pi R\).
- Substitute the values: \(B = \frac{\mu_0 I}{4\pi R^2} (2\pi R) = \frac{\mu_0 I}{2R}\).
- Calculate: \(B = \frac{(4\pi \times 10^{-7}) (5.0)}{2(0.10)} = \frac{20\pi \times 10^{-7}}{0.20} = 100\pi \times 10^{-7} \approx 3.14 \times 10^{-5}\ \mathrm{T}\).
Units check. \(\frac{\mathrm{T\cdot m/A} \cdot \mathrm{A}}{\mathrm{m}} = \mathrm{T}\). The units are consistent.
Three quick tests
- What is the direction of the magnetic field at a point located on the axis of a circular current loop, far from the center?
Answer
Along the axis, the field points in the direction of the magnetic dipole moment, determined by the right-hand rule for the current circulation. - Why does the magnetic field from a straight wire decrease as \(1/r\) rather than \(1/r^2\)?
Answer
Although each element contributes as \(1/r^2\), the integration over the infinite length of the wire introduces a factor of \(r\) from the angular limits, resulting in a net \(1/r\) dependence. - Can the Biot–Savart law be used to find the field of a single electron moving in a straight line?
Answer
Only approximately. For a slow charge (\(v \ll c\)) the field is close to \(\vec B \approx \frac{\mu_0}{4\pi}\frac{q\,\vec v \times \hat s}{s^2}\), but a single moving charge is not a steady current; the exact field needs the Liénard–Wiechert (retarded) fields.
Problem set for PU-102 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.