AcademySources of Magnetic Fields
Academy
Fields from Current Loops
Level 1 - Physics topic page in Sources of Magnetic Fields.
Principle
A current loop produces a magnetic field like a magnetic dipole. On the loop axis, the field points along the axis and can be found by adding Biot-Savart contributions.
Notation
\(I\)
current in the loop
\(\mathrm{A}\)
\(R\)
loop radius
\(\mathrm{m}\)
\(x\)
distance along the loop axis from the centre
\(\mathrm{m}\)
\(N\)
number of turns
\(A\)
area of one loop
\(\mathrm{m^{2}}\)
\(\vec m\)
magnetic dipole moment
\(\mathrm{A\,m^{2}}\)
Method
Derivation 1: Use axial symmetry
For a circular loop, transverse field components from opposite current elements cancel on the axis. Axial components add.
Axial field of one loop
\[B(x)=\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}\]
Field at the centre
\[B(0)=\frac{\mu_0 I}{2R}\]
N turns
\[B_N=N B_{\mathrm{one\ loop}}\]
Derivation 2: Assign the magnetic moment
The direction of the magnetic moment is given by the right-hand rule: curl fingers with the current and the thumb points along \(\vec m\).
Magnetic dipole moment
\[\vec m=NIA\hat n\]
Derivation 3: Interpret the far field
Far from the loop, the field resembles that of a bar magnet: one side behaves like a north pole and the other like a south pole. The loop is not a true isolated pole; field lines remain closed loops.
Rules
Loop axis field
\[B(x)=\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}\]
Loop centre
\[B=\frac{\mu_0 I}{2R}\]
Magnetic dipole moment
\[\vec m=NIA\hat n\]
Examples
Question
A circular loop carries counterclockwise current as viewed by you. Does the magnetic field at the centre point toward you or away from you?
Answer
Curl the fingers of your right hand counterclockwise. Your thumb points toward you, so the field at the centre points toward you.
Checks
- At the loop centre, a smaller radius gives a stronger field for the same current.
- Multiple tightly wound turns multiply the field approximately by \(N\).
- The magnetic moment points normal to the loop.
- A current loop is a dipole, not an isolated magnetic pole.