91Ó°ÊÓ

In the context of magnetism, stable equilibrium occurs when a magnetic dipole is aligned opposite to the direction of an external uniform magnetic field. This alignment minimizes the system's total potential energy, meaning any small disturbance still keeps it close to this equilibrium and the magnet tends to return to this position.
A key aspect of stable equilibrium is its tendency to resist changes. A small disturbance results in forces or torques that act to return the object to its original position.

When calculating the potential energy (\( U \)) of a magnetic dipole in a magnetic field, it is given by the formula: \[ U = -m \cdot B \cdot \cos(\theta)\] where:

• \( m \) is the magnetic moment of the dipole,
• \( B \) is the magnetic field strength,
• \( \theta \) is the angle between the magnetic moment and the magnetic field direction.

For stable equilibrium, this angle \( \theta \) is \( 0 \), hence cosine of \( 0 \) is 1, making the potential energy: \[ U = -m \cdot B \] in this particular scenario.
In our given example, calculating with values \( m=0.32 \, J/T \) and \( B=0.15 \, T \), the result is \( U = -0.048 \, J \), shedding light on how the system has minimized energy at stable equilibrium.

Unstable equilibrium, in magnetism, refers to the scenario where the magnetic dipole is aligned in the same direction as the external magnetic field. This represents a state of maximized potential energy, where even a slight disturbance can change the magnet's position significantly, causing it to move away from equilibrium.
In this situation, the angle between the magnetic moment and the magnetic field, \( \theta \), is \( 180^\circ \) or \( \pi \) radians. The cosine of \( \theta \) here is -1, turning the potential energy of the system into:\[ U = -m \cdot B \cdot \cos(\pi) = m \cdot B \]

• This indicates that the system has the highest potential energy, indicative of an unstable state.

In our example, using given values \( m=0.32 \, J/T \) and \( B=0.15 \, T \), the calculation results in \( U = 0.048 \, J \). The positive value signifies the high energy state which the system tends to move away from when disturbed.

Potential energy in the context of magnetism describes the energy stored due to the position of a magnetic dipole within a magnetic field. It depends on the orientation of the magnetic moment relative to the magnetic field. Simply put, this is the energy that can potentially do work when the dipole reorientates.
The mathematical formula is given by: \[ U = -m \cdot B \cdot \cos(\theta) \] When \( \theta \) is 0 (stable equilibrium), potential energy is minimized. Conversely, when \( \theta \) is \( 180^\circ \) or \( \pi \) (unstable equilibrium), it reaches a maximum level.
This potential energy can be visualized as:

• In stable equilibrium, the magnetic dipole feels a torque resulting in a re-alignment to lower the energy further.
• In unstable equilibrium, the slightest rotation results in a re-alignment to drastically lower energy making this state less likely to persist spontaneously.