Problem 59 - University Physics with Modern Physics

Question

Torque on a Dipole. An electric dipole with dipole moment \(\vec{p}\) is in a uniform electric field \(E\) . (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 displacement 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) Torque is zero when the dipole is parallel or anti-parallel to the field. (b) Parallel is stable; anti-parallel is unstable. (c) In stable alignment, the dipole's field opposes the external field.

Step 1: Understanding Torque on a Dipole

The torque \( \tau \) on a dipole in an electric field can be expressed as \( \tau = \vec{p} \times \vec{E} \). This formula shows that torque depends on the angle between \( \vec{p} \) and \( \vec{E} \) due to the cross product nature, specifically on \( \sin(\theta) \), where \( \theta \) is the angle between \( \vec{p} \) and \( \vec{E} \).

Step 2: Condition for Zero Torque

For the torque to be zero, \[ \tau = \vec{p} \times \vec{E} = \|\vec{p}\| \|\vec{E}\| \sin(\theta) = 0 \]The condition \( \sin(\theta) = 0 \) indicates that \( \theta \) must be 0° or 180°, meaning the dipole must be aligned parallel or anti-parallel to the electric field.

Step 3: Evaluating Stability

For a stable orientation, consider a small angular displacement \( \Delta\theta \) from equilibrium. If \( \theta = 0^{\circ} \) (dipole aligned parallel to \( \vec{E} \)), a restoring torque acts to return the dipole to this position, indicating stability. If \( \theta = 180^{\circ} \) (dipole anti-parallel to \( \vec{E} \)), the torque increases the displacement, indicating instability.

Step 4: Understanding Dipole's Own Field Effect in Stable Orientation

In the stable orientation (\( \theta = 0^{\circ} \), the dipole's electric field points in the opposite direction to the external electric field due to the nature of dipoles to align themselves to minimize energy. Thus, the dipole's field tends to oppose the external field, as this arrangement leads to stable equilibrium.

Key Concepts

Torque on a Dipole

When considering an electric dipole in a uniform electric field, torque plays a crucial role in determining its behavior. Torque, denoted as \( \tau \), helps us understand how a dipole tends to rotate in the presence of an electric field. The torque on a dipole can be expressed by the formula \( \tau = \vec{p} \times \vec{E} \). Here, \( \vec{p} \) is the dipole moment, which characterizes the strength and direction of the dipole, while \( \vec{E} \) represents the electric field. This equation relies on the cross product, indicating that the torque depends on the sine of the angle \( \theta \) between the dipole moment and the electric field.

If the dipole moment is aligned parallel (\( \theta = 0^{\circ} \)) or anti-parallel (\( \theta = 180^{\circ} \)) to the electric field, the torque on the dipole is zero. This is because the sine of these angles is zero, making the product zero.

Electric Field

An electric field is a force field that surrounds electric charges. It influences other charges in its vicinity, causing them to experience a force. In the context of a dipole, the electric field exerts a pivotal role in aligning the dipole to a position of either minimal or maximal potential energy.

The electric field is uniform when its magnitude and direction remain constant across the space it occupies. When a dipole is placed in such a uniform field, the uniformity ensures that the torques are influenced mainly by the angles of the alignment without any spatial variation in the field.

Stability of Dipoles

The stability of a dipole in an electric field refers to how it returns to equilibrium after being disturbed. To determine stability, consider small angular displacements from equilibrium positions. If a restoring torque brings the dipole back to its alignment, it is considered stable. In this scenario, when \( \theta = 0^{\circ} \), any deviation results in a torque that works diligently to realign the dipole with the field.

When \( \theta = 180^{\circ} \), if the dipole experiences a small displacement, the torque exacerbates the displacement further, pushing it away from alignment. Therefore, this position is unstable and does not bring the dipole back to its original orientation.

Dipole Moment

The dipole moment \( \vec{p} \) is a vector quantity that represents the electric strength and the direction of the dipole. It is defined as the product of the charge magnitude \( q \) and the distance \( d \) between the charges: \( \vec{p} = q \cdot d \).

This vector points from the negative to the positive charge, providing directionality to the dipole. In equilibrium alignment, the dipole moment is crucial, especially in dictating the behavior of the dipole in an electric field. For example, when aligned parallel to the electric field, the dipole is positioned to minimize its potential energy, thereby leading to a stable equilibrium.