Problem 55 - University Physics with Modern Physics

Question

An electric dipole with dipole moment \(\overrightarrow{p}\) is in a uniform external electric field \(\overrightarrow{E}\) . (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (\(Hint:\) Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Step-by-Step Solution

Answer

(a) Zero torque at 0° and 180°. (b) 0° is stable, 180° is unstable. (c) Dipole's field aligns with the external field, minimizing it at stability.

Step 1: Understand the Torque on a Dipole

The torque \( \tau \) on an electric dipole in an external electric field \( \overrightarrow{E} \) is given by the cross product \( \overrightarrow{\tau} = \overrightarrow{p} \times \overrightarrow{E} \), where \( \overrightarrow{p} \) is the dipole moment. For the torque to be zero, \( \overrightarrow{p} \) must be either parallel or anti-parallel to \( \overrightarrow{E} \), making the angle \( \theta \) between them equal to 0 or 180 degrees.

Step 2: Determine Equilibrium Orientations

The torque on the dipole is zero when the angle \( \theta \) is either 0 degrees or 180 degrees. Thus, there are two equilibrium orientations: when the dipole moment \( \overrightarrow{p} \) is aligned with \( \overrightarrow{E} \) (\( \theta = 0 \)) and when it is opposite (\( \theta = 180 \)).

Step 3: Identify Stable and Unstable Equilibria

To determine stability, consider a small perturbation in \( \theta \). If the dipole returns to its initial orientation, the equilibrium is stable. If \( \theta = 0 \) (dipole parallel to the field), any small rotation increases potential energy, leading the dipole back, indicating stability. Conversely, if \( \theta = 180 \) (dipole anti-parallel), perturbation decreases potential energy, enhancing rotation, indicating instability.

Step 4: Rationalize Dipole's Interaction with the Field

In the stable equilibrium at \( \theta = 0 \), the dipole aligns with the field, minimizing potential energy. A small rotation resists change due to restoring forces, demonstrating stability. Therefore, in this position, the dipole moment is aligned such that its own electric field reduces the influence of the external field.

Key Concepts

Torque on a Dipole

When we talk about a dipole in an external electric field, it's crucial to understand the concept of torque. Torque, in this context, is like a twisting force that acts on the electric dipole. This force tries to align the dipole with the electric field. Mathematically, the torque \( \tau \) on a dipole with a dipole moment \( \overrightarrow{p} \) in a uniform electric field \( \overrightarrow{E} \) is calculated using the cross product: \( \overrightarrow{\tau} = \overrightarrow{p} \times \overrightarrow{E} \).

• If the dipole moment is perfectly aligned with the electric field (angle \( \theta \) is 0 degrees), or exactly opposite to it (angle \( \theta \) is 180 degrees), the torque becomes zero.
• This is because the sine component of the angle in the cross product goes to zero, eliminating the twisting effect.

Understanding this zero torque condition is key, because it identifies the specific angular positions where the dipole does not experience rotational forces.

Equilibrium Orientations

Equilibrium orientations describe the positions where the dipole does not rotate further in the electric field. We determined that these occur when the torque is zero.

• When the dipole moment \( \overrightarrow{p} \) is aligned in the same direction as the electric field \( \overrightarrow{E} \) (\( \theta = 0 \)), or
• When \( \overrightarrow{p} \) is exactly opposite to \( \overrightarrow{E} \) (\( \theta = 180 \)).

These orientations are crucial because they help predict the behavior of the dipole under the influence of the field. Equilibrium can be explained simply as the 'resting' positions of the dipole where no net rotations are happening. This concept ties closely with stability, as we will see next.

Stable and Unstable Equilibria

In physics, stability is about how a system behaves after a slight disturbance. For a dipole, this means slightly rotating it from its equilibrium position and observing the result.

• Stable Equilibrium: If the dipole is initially parallel to the field (\( \theta = 0 \)), any small rotation will increase the dipole's potential energy. This increase drives it back to its starting orientation. That's why it's considered stable; it naturally returns to this orientation.
• Unstable Equilibrium: Contrarily, when the dipole is anti-parallel (\( \theta = 180 \)), a small turn will decrease its potential energy further, causing it to continue rotating away from this position. Hence, anti-parallel alignment is deemed unstable.

In essence, these observations reveal how the dipole seeks configurations of lower potential energy, which helps to understand broader phenomena like molecular alignments in external fields or even more complex electrostatic applications. Understanding these distinctions ensures clarity when predicting dipole behavior!