Additional: Magnetic Potential Energy of a Dipole

Potential Energy of a Magnetic Dipole in a Magnetic Field ($ U = -\vec{M} \cdot \vec{B} $)

When a magnetic dipole, such as a current loop or a bar magnet, is placed in an external magnetic field, it experiences a torque that tends to align its magnetic dipole moment with the field. To rotate the dipole against this torque, work must be done by an external agent. This work done is stored as the magnetic potential energy of the dipole in the magnetic field.

This concept is analogous to the electric potential energy of an electric dipole in an electric field ($U = -\vec{p} \cdot \vec{E}$) or the gravitational potential energy of an object in a gravitational field (where the force tries to move the object towards lower potential).

---

Derivation of Magnetic Potential Energy

Consider a magnetic dipole with magnetic dipole moment $\vec{M}$ placed in a uniform external magnetic field $\vec{B}$. The torque exerted on the dipole by the magnetic field is given by:

$ \vec{\tau} = \vec{M} \times \vec{B} $

The magnitude of this torque is $\tau = MB \sin\theta$, where $\theta$ is the angle between $\vec{M}$ and $\vec{B}$. This torque acts to reduce the angle $\theta$.

Suppose we want to rotate the dipole from an initial angle $\theta_0$ to a final angle $\theta_f$ with respect to the magnetic field. An external torque $\vec{\tau}_{ext}$ must be applied to counteract the magnetic torque $\vec{\tau}$. To rotate the dipole slowly (without changing kinetic energy), the external torque must be equal and opposite to the magnetic torque: $\vec{\tau}_{ext} = -\vec{\tau} = -(\vec{M} \times \vec{B})$. The magnitude of the external torque is $\tau_{ext} = MB \sin\theta$.

The work done by the external torque in rotating the dipole through an infinitesimal angular displacement $d\theta$ is $dW = \tau_{ext} \, d\theta$. Assuming the rotation is around an axis perpendicular to both $\vec{M}$ and $\vec{B}$ (initially), or more generally, that $d\vec{\theta}$ is parallel to $\vec{\tau}_{ext}$:

$ dW = (MB \sin\theta) \, d\theta $

The total work done by the external agent in rotating the dipole from $\theta_0$ to $\theta_f$ is:

$ W = \int_{\theta_0}^{\theta_f} MB \sin\theta \, d\theta $

$ W = MB \int_{\theta_0}^{\theta_f} \sin\theta \, d\theta $

$ W = MB [-\cos\theta]_{\theta_0}^{\theta_f} $

$ W = MB (-\cos\theta_f - (-\cos\theta_0)) $

$ W = MB (\cos\theta_0 - \cos\theta_f) $

This work done is stored as the change in potential energy of the dipole: $\Delta U = W$.

$ U(\theta_f) - U(\theta_0) = MB (\cos\theta_0 - \cos\theta_f) $

To define the potential energy $U(\theta)$ at an arbitrary angle $\theta$, we need to choose a reference point where the potential energy is zero. A convenient reference point is when the dipole moment is perpendicular to the magnetic field ($\theta_0 = 90^\circ$), where the torque is maximum, but the dot product is zero. Let $U(90^\circ) = 0$.

Setting $\theta_0 = 90^\circ$ and $U(90^\circ) = 0$:

$ U(\theta_f) - 0 = MB (\cos 90^\circ - \cos\theta_f) $

$ U(\theta_f) = MB (0 - \cos\theta_f) = -MB \cos\theta_f $

Replacing $\theta_f$ with the general angle $\theta$, the potential energy of a magnetic dipole $\vec{M}$ in a magnetic field $\vec{B}$ is:

$ U(\theta) = -MB \cos\theta $

This expression can be written concisely using the dot product of the magnetic dipole moment vector $\vec{M}$ and the magnetic field vector $\vec{B}$:

$ U = -\vec{M} \cdot \vec{B} $

---

Physical Interpretation of Potential Energy

The potential energy $U = -MB \cos\theta$ depends on the orientation of the magnetic dipole moment vector $\vec{M}$ relative to the magnetic field $\vec{B}$.

• Minimum Potential Energy (Stable Equilibrium): The potential energy is minimum when $\cos\theta$ is maximum, i.e., $\cos\theta = 1$, which means $\theta = 0^\circ$.

  $ U_{min} = -MB \cos 0^\circ = -MB $

  This occurs when the magnetic dipole moment $\vec{M}$ is aligned parallel to the magnetic field $\vec{B}$. This is a state of stable equilibrium, as the torque is zero ($\tau = MB \sin 0^\circ = 0$) and any small displacement away from this position results in a torque that tends to restore the dipole to this alignment. Magnets tend to align themselves with the external magnetic field in this way.
• Maximum Potential Energy (Unstable Equilibrium): The potential energy is maximum when $\cos\theta$ is minimum, i.e., $\cos\theta = -1$, which means $\theta = 180^\circ$.

  $ U_{max} = -MB \cos 180^\circ = -MB (-1) = +MB $

  This occurs when the magnetic dipole moment $\vec{M}$ is aligned anti-parallel to the magnetic field $\vec{B}$. This is a state of unstable equilibrium, as the torque is still zero ($\tau = MB \sin 180^\circ = 0$), but any small displacement results in a torque that tends to move the dipole away from this alignment towards the stable equilibrium.
• Zero Potential Energy (Reference Point): As per our choice of reference, potential energy is zero when $\theta = 90^\circ$ (or $270^\circ$).

  $ U(90^\circ) = -MB \cos 90^\circ = 0 $

  This occurs when the magnetic dipole moment $\vec{M}$ is perpendicular to the magnetic field $\vec{B}$.

The potential energy concept helps in understanding how magnetic dipoles behave in magnetic fields and is crucial in areas like the study of magnetic materials and nuclear magnetic resonance (NMR).

---

Answer:

---

Answer: