📂Electrodynamics

The Torque on a Magnetic Dipole in an External Magnetic Field and Paramagnetism

Explanation¹

Just like an electric dipole gains torque by an external electric field, a magnetic dipole does the same. Let’s assume there is a current loop in a uniform external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$ as shown below. Since a small rectangular current loop can be overlapped to approximate a current loop of any shape, let’s focus on the rectangular current loop. The magnetic force received by each side can be calculated using the Lorentz law. Naturally, the forces on sides 2 and 4 have the same magnitude, and their directions are opposite to each other as per the right-hand rule. Thus, the forces on sides 2 and 4 cancel each other out. The forces on sides 1 and 3 can be decomposed as shown in the figure above. Therefore, it can be seen that the current loop rotates and torque can be calculated.

$$ \mathbf{N}=\dfrac{1}{2}aF\sin \theta \hat{\mathbf{x}}+\dfrac{1}{2}aF\sin\theta \hat{\mathbf{x}}=aF\sin\theta \hat{\mathbf{x}} $$

The magnitude of the magnetic force is

$$ |\mathbf{F}|=\left| I \int (d\mathbf{l} \times \mathbf{B} ) \right| = IbB $$

Therefore, the torque is

$$ \mathbf{N}=IabB\sin\theta \hat{\mathbf{x}}=mB\sin\theta\hat{\mathbf{x}} \\ \implies \mathbf{N}=\mathbf{m}\times\mathbf{B} $$

$m=Iab$ represents the magnitude of the magnetic dipole moment of the loop.

This is the torque acting on a current loop in a uniform magnetic field, and it must be calculated differently when the magnetic field is not uniform. The result above is similar to the form of the torque on an electric dipole caused by an external electric field $\mathbf{N}=\mathbf{p} \times \mathbf{E}$. Due to the torque mentioned above, the current loop rotates until it becomes parallel to the $xy$ plane, making the direction of the dipole moment align with the direction of the external magnetic field. This is how diamagnetism occurs.

Since all electrons possess a magnetic dipole, diamagnetism can be considered a general phenomenon; however, it is not. According to Pauli’s exclusion principle, electrons inside an atom pair up with another electron that has an opposite spin direction. Therefore, diamagnetism appears in atoms or molecules with an odd number of electrons and is caused by the unpaired electron receiving rotational force. The force received by a very small loop with a magnetic dipole moment of $\mathbf{m}$ in a non-uniform magnetic field $\mathbf{B}$ is as follows:

$$ \mathbf{F}=\nabla (\mathbf{m} \cdot \mathbf{B}) $$

This is also similar to the force experienced by an electric dipole in an electric field $\mathbf{F}=\nabla(\mathbf{p} \cdot \mathbf{E})$.