AcademySources of Magnetic Fields
Academy
Ampere's Law
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
Ampere's law relates the circulation of the magnetic field around a closed path to the current enclosed by that path.
Notation
\(\oint \vec B\cdot d\vec l\)
magnetic circulation around a closed path
\(\mathrm{T\,m}\)
\(I_{\mathrm{enc}}\)
net current through the surface bounded by the path
\(\mathrm{A}\)
\(d\vec l\)
small path element in the chosen loop direction
\(\mathrm{m}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
Method
Derivation 1: State the law
Choose a closed path and a direction around it. The enclosed current is signed by the right-hand rule relative to that path direction.
Ampere's law
\[\oint \vec B\cdot d\vec l=\mu_0 I_{\mathrm{enc}}\]
Derivation 2: Choose a useful Amperian loop
Ampere's law is most useful when symmetry makes \(\vec B\) tangent to the loop and constant in magnitude along parts of it.
Constant tangent field
\[\oint \vec B\cdot d\vec l=B\oint dl\]
Circular path
\[\oint dl=2\pi r\]
Derivation 3: Recover the straight-wire field
For a long straight wire, a circular loop of radius \(r\) encloses current \(I\). The field is tangent and constant on the circle.
Apply Ampere's law
\[B(2\pi r)=\mu_0 I\]
Straight-wire result
\[B=\frac{\mu_0 I}{2\pi r}\]
Rules
Ampere's law
\[\oint \vec B\cdot d\vec l=\mu_0 I_{\mathrm{enc}}\]
Signed current
\[I_{\mathrm{enc}}=\sum I_{\mathrm{through\ surface}}\]
Best use case
\[\vec B\cdot d\vec l=B\,dl\quad\text{when symmetry allows}\]
Examples
Question
A circular Amperian loop surrounds a long wire carrying
\[6.0\,\mathrm A\]
What is \[\oint\vec B\cdot d\vec l\]
?Answer
\[\oint\vec B\cdot d\vec l=\mu_0 I_{\mathrm{enc}}=(4\pi\times10^{-7})(6.0)=7.5\times10^{-6}\,\mathrm{T\,m}\]
Checks
- The path is closed.
- The current must pierce a surface bounded by the path.
- Currents are signed by the right-hand rule.
- Ampere's law is powerful for high-symmetry fields, not a universal shortcut for every geometry.