AcademyMagnetic Fields and Forces
Academy
Torque on Current Loops
Level 1 - Physics topic page in Magnetic Fields and Forces.
Principle
A current loop in a magnetic field experiences torque that aligns its magnetic moment.
Notation
\(\vec\mu\)
magnetic dipole moment of a loop
\(\mathrm{A\,m^{2}}\)
\(N\)
number of turns
\(I\)
loop current
\(\mathrm{A}\)
\(A\)
area of one loop
\(\mathrm{m^{2}}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(\tau\)
torque magnitude
\(\mathrm{N\,m}\)
Method
For a flat loop, opposite sides feel equal and opposite magnetic forces. They make a couple rather than a net force.
Dipole moment
\[\mu=NIA\]
Torque vector
\[\vec\tau=\vec\mu\times\vec B\]
Torque magnitude
\[\tau=NIAB\sin\theta\]
Potential energy
\[U=-\vec\mu\cdot\vec B\]
The direction of \(\vec\mu\) is set by a right-hand rule: curl fingers with conventional current; thumb points along \(\vec\mu\).
Rules
Magnetic moment
\[\mu=NIA\]
Loop torque
\[\vec\tau=\vec\mu\times\vec B\]
Torque magnitude
\[\tau=NIAB\sin\theta\]
Dipole energy
\[U=-\vec\mu\cdot\vec B\]
Examples
Question
A
\[50\]
-turn coil has area \[2.0\times10^{-3}\,\mathrm{m^2}\]
current \[0.40\,\mathrm{A}\]
and is in \[0.30\,\mathrm{T}\]
Find maximum torque.Answer
\[\tau_{\max}=NIAB=(50)(0.40)(2.0\times10^{-3})(0.30)=1.2\times10^{-2}\,\mathrm{N\,m}\]
Checks
- Area vector and magnetic moment use the right-hand rule for current.
- Maximum torque occurs when \(\vec\mu\) is perpendicular to \(\vec B\).
- Stable alignment has \(\vec\mu\) parallel to \(\vec B\).