What is the work done in rotating a dipole from its unstable equilibrium to stable equilibrium? Does the energy of the dipole increase or decrease?

To understand the work done in rotating a dipole from an unstable equilibrium to a stable equilibrium, we first need to clarify what we mean by a dipole and the concepts of equilibrium. A dipole consists of two equal and opposite charges separated by a distance, creating an electric field. The dipole moment is a vector quantity that points from the negative charge to the positive charge. When we talk about equilibrium, we refer to the positions where the net forces acting on the dipole are balanced.

Equilibrium Positions of a Dipole

There are two types of equilibrium for a dipole in an external electric field:

• Stable Equilibrium: This occurs when the dipole is aligned with the electric field. In this position, any small displacement will result in a restoring torque that brings the dipole back to its original position.
• Unstable Equilibrium: This is when the dipole is oriented opposite to the electric field. Here, any small displacement will cause the dipole to move further away from this position, as the torque acts in the direction of the field.

Work Done in the Rotation

When you rotate the dipole from the unstable equilibrium (180 degrees from the field direction) to the stable equilibrium (0 degrees with respect to the field), you are essentially moving it from a position of higher potential energy to a position of lower potential energy. The work done on the dipole can be calculated using the formula:

W = -ΔU

Where:

• W is the work done on the dipole.
• ΔU is the change in potential energy.

In this case, as you rotate the dipole, you are decreasing its potential energy because it is moving to a more stable configuration. The potential energy (U) of a dipole in an electric field (E) is given by:

U = -p · E

Where:

• p is the dipole moment vector.
• E is the electric field vector.

At unstable equilibrium, the angle θ between the dipole moment and the electric field is 180 degrees, resulting in maximum potential energy. At stable equilibrium, the angle is 0 degrees, leading to minimum potential energy.

Energy Considerations

As the dipole rotates from the unstable to the stable position, the potential energy decreases, which means that the energy of the dipole decreases. The work done by the external agent to rotate the dipole is negative, indicating that energy is released as the dipole moves to a more stable configuration. This energy can be thought of as being converted into kinetic energy if the dipole is allowed to move freely.

Summary of Key Points

• Work is done when rotating a dipole from unstable to stable equilibrium.
• The potential energy of the dipole decreases during this rotation.
• The energy of the dipole decreases, indicating a transition to a more stable state.

In essence, the work done in this process reflects the fundamental principles of energy conservation and stability in electric fields. The dipole naturally seeks the lowest energy state, which is why it moves towards stable equilibrium when given the chance.