Question

Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state? a) \(0 \mathrm{rad}\) d) The electric dipole moment is b) \(\pi / 2\) rad not stable under any condition in c) \(\pi\) rad an applied electric field.

Short answer

Answer: a) \(0 \mathrm{rad}\)

Step-by-step solution

Understand the electric dipole moment and potential energy

In a system with an electric dipole moment and an applied electric field, the potential energy (U) of the system can be described as: $$ U = -\vec{p} \cdot \vec{E} $$ where \(\vec{p}\) is the electric dipole moment and \(\vec{E}\) is the applied electric field. The dot product between the two vectors depends on the angle between them, denoted as \(\theta\). Therefore, the potential energy formula can be rewritten as: $$ U = -p E \cos{\theta} $$

Analyze the potential energy with respect to the angle θ

We need to determine which angle θ between the electric dipole moment and applied electric field results in the most stable state. The most stable state corresponds to the lowest potential energy. We can examine the given options one by one. Let's evaluate the potential energy for each given angle: 1. Angle θ = \(0 \mathrm{rad}\): $$ U_{0} = -p E \cos{(0 \mathrm{rad})} = -p E $$ 2. Angle θ = \(\pi / 2\) rad: $$ U_{\pi/2} = -p E \cos{(\pi / 2 \mathrm{rad})} = 0 $$ 3. Angle θ = \(\pi\) rad: $$ U_{\pi} = -p E \cos{(\pi \mathrm{rad})} = p E $$

Determine the angle corresponding to the most stable state

To find the most stable state, we need to find the angle that results in the lowest potential energy. Comparing the values of the potential energy calculated in step 2, we can see that: - \(U_{0}\) = \(-p E\): The potential energy is negative, indicating a stable state. - \(U_{\pi/2}\) = \(0\): The potential energy is equal to zero, indicating a less stable state than when the potential energy is negative. - \(U_{\pi}\) = \(p E\): The potential energy is positive, indicating an unstable state. Therefore, the angle that results in the most stable state is \(0 \mathrm{rad}\), as it leads to the lowest potential energy. Hence, the correct answer is: a) \(0 \mathrm{rad}\)

Key concepts

Understanding Potential Energy in Electric Fields

Potential energy is a key concept in physics, especially when discussing electric dipoles within an electric field. An electric dipole consists of two opposite charges separated by a distance. The potential energy, denoted as \( U \), in the presence of an applied electric field \( \vec{E} \), can be defined as:\[ U = -\vec{p} \cdot \vec{E} \]where \( \vec{p} \) is the electric dipole moment.

The formula simplifies to:\[ U = - p E \cos{\theta} \]\( \theta \) is the angle between the dipole moment and the electric field vector.

• When \( \theta = 0 \), \( \cos{\theta} = 1 \), resulting in \( U = -pE \), the lowest potential energy, indicating highest stability.
• When \( \theta = \pi/2 \), \( \cos{\theta} = 0 \), resulting in \( U = 0 \), meaning less stability compared to when the potential energy is negative.
• When \( \theta = \pi \), \( \cos{\theta} = -1 \), leading to \( U = pE \), which is the highest potential energy, indicative of an unstable state.

These principles help in determining the stability of an electric dipole in an applied field.

Role of Applied Electric Field

An applied electric field influences the orientation and behavior of electric dipoles. The external field initiates forces on the dipole, causing it to align itself in the direction of the field.

The electric field \( \vec{E} \) interacts with the dipole moment \( \vec{p} \), producing a torque that tends to rotate the dipole to minimize potential energy.

• At \( \theta = 0 \), the dipole is aligned with the field, maximizing the effectiveness of this alignment.
• At \( \theta = \pi/2 \), the dipole is perpendicular to the field, minimizing interaction.
• At \( \theta = \pi \), the dipole is opposite to the field, leading to maximum potential energy and instability.

Understanding how an electric field acts on a dipole helps predict and control system behaviors, crucial in fields like chemistry and electronics.

Stability in Electric Fields

Stability in electric fields is all about the orientation of the electric dipole relative to the field. A stable configuration minimizes potential energy, resulting in less movement or disturbance in the system.

For an electric dipole in a field:

• Stable State (\( \theta = 0 \)): The dipole aligns with the field, meaning less potential for rotational motion and energy loss. This alignment maintains the system's equilibrium.
• Neutral State (\( \theta = \pi/2 \)): The dipole is perpendicular. Here, the forces are balanced, but it's neither fully stable nor unstable.
• Unstable State (\( \theta = \pi \)): The dipole opposes the field direction, attempting to realign when disturbed since any external force can cause movement.

The concept of stability is vital in applications like molecular interactions and designing stable electric components, where unwanted motion can disrupt performance.