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.