AcademySources of Magnetic Fields
Academy
Biot-Savart for Current Elements
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
The Biot-Savart law adds the magnetic fields made by many small current elements. It is the current version of the moving-charge field law.
Notation
\(I\)
current in the wire
\(\mathrm{A}\)
\(d\vec l\)
small length vector in the current direction
\(\mathrm{m}\)
\(\vec r\)
vector from current element to field point
\(\mathrm{m}\)
\(\hat r\)
unit vector from element to field point
\(d\vec B\)
small magnetic field contribution
\(\mathrm{T}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
Method
Derivation 1: Replace moving charges with current elements
A steady current represents many moving charges. For a short element of wire, the source strength is \(I\,d\vec l\), directed along conventional current.
Current element
\[I\,d\vec l\]
Biot-Savart law
\[d\vec B=\frac{\mu_0}{4\pi}\frac{I\,d\vec l\times\hat r}{r^2}\]
Magnitude
\[dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2}\]
Derivation 2: Integrate over the source
The total magnetic field is the vector sum of all contributions. Symmetry can make this easier by causing components to cancel.
Vector sum
\[\vec B=\int d\vec B\]
Full law
\[\vec B=\frac{\mu_0 I}{4\pi}\int\frac{d\vec l\times\hat r}{r^2}\]
Derivation 3: Track direction before magnitude
The cross product direction changes from element to element. Decide which components survive before doing the integral.
Rules
Biot-Savart law
\[d\vec B=\frac{\mu_0}{4\pi}\frac{I\,d\vec l\times\hat r}{r^2}\]
Integrated field
\[\vec B=\frac{\mu_0 I}{4\pi}\int\frac{d\vec l\times\hat r}{r^2}\]
Contribution magnitude
\[dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2}\]
Examples
Question
A current element points to the right. The field point is directly above it. What is the direction of \(d\vec B\)?
Answer
Use \(d\vec l\times\hat r\). Right crossed with up points out of the page, so \(d\vec B\) is out of the page.
Checks
- \(d\vec l\) points along conventional current.
- Each contribution is perpendicular to both \(d\vec l\) and \(\hat r\).
- Integrate vectors, not just magnitudes.
- Biot-Savart is usually best when Ampere's law has insufficient symmetry.