Magnetic Dipole Moment Unit

The magnetic dipole moment is a fundamental property of a magnetic dipole, analogous to the electric dipole moment. It is a vector quantity that represents the strength and orientation of a magnetic source.

The magnetic dipole moment ($\vec{m}$) of a current loop carrying current $I$ and having area $A$ is defined as:

\(\vec{m} = I\vec{A}\)

Let's determine the unit of magnetic dipole moment based on this definition:

• The unit of current ($I$) is Ampere (A).
• The unit of area ($A$) is metre squared (m\({}^2\)).

Therefore, the unit of magnetic dipole moment is the product of the unit of current and the unit of area:

Unit of \(\vec{m}\) = Unit of $I$ \(\times\) Unit of $\vec{A}$

Unit of \(\vec{m}\) = Ampere \(\times\) metre\({}^2\)

Unit of \(\vec{m}\) = Ampere-metre\({}^2\)

Another way to relate magnetic dipole moment is through the torque ($\vec{\tau}$) experienced by a magnetic dipole in a magnetic field ($\vec{B}$):

\(\vec{\tau} = \vec{m} \times \vec{B}\)

The unit of torque is Newton-metre (N m). The unit of magnetic field can be Tesla (T), which is equivalent to Newton per Ampere-metre (N/(A m)).

From \(\tau = mB\sin\theta\), the unit of $m$ can be found as:

Unit of $m$ = Unit of \(\tau\) / Unit of $B$

Unit of $m$ = (N m) / (N/(A m))

Unit of $m$ = (N m) \(\times\) (A m / N)

Unit of $m$ = A m\({}^2\)

Both definitions lead to the same unit for magnetic dipole moment: Ampere-metre\({}^2\).

Let's examine the given options:

• Option 1: newton/(ampere-metre) (N/(A m)) is the unit of magnetic field (Tesla).
• Option 2: ampere/metre (A/m) is related to magnetic field strength or magnetization, but not the magnetic dipole moment itself.
• Option 3: ampere-metre\({}^2\) (A m\({}^2\)) matches the unit derived from the definition.

Based on the analysis, the unit of magnetic dipole moment is ampere-metre\({}^2\).