Question

What is the angle between the electric dipole moment \(\vec{p}\) and the electric field strength \(\vec{E}\) when the dipole is in a stable equilibrium?

• A. \(\frac{\pi}{4}\)
• B. \(\pi\)
• C. \(\frac{\pi}{2}\)
• D. 0

Hint:

Think of a magnetic compass in the Earth's magnetic field. It's in stable equilibrium when the north pole of the compass needle points towards the Earth's magnetic north pole (i.e., they are aligned). An electric dipole in an electric field behaves similarly. Stable = Aligned (\(0^\circ\)), Unstable = Anti-aligned (\(180^\circ\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
An electric dipole placed in a uniform electric field experiences a torque that tends to align it with the field. The potential energy of the dipole depends on its orientation with respect to the field. Equilibrium occurs when the torque is zero, and stable equilibrium corresponds to the orientation with the minimum potential energy.
Step 2: Key Formula or Approach:
The torque (\(\vec{\tau}\)) on a dipole is given by \(\vec{\tau} = \vec{p} \times \vec{E}\), with magnitude \(\tau = pE \sin\theta\).
The potential energy (\(U\)) of the dipole is given by \(U = -\vec{p} \cdot \vec{E}\), with magnitude \(U = -pE \cos\theta\).
Here, \(\theta\) is the angle between the dipole moment \(\vec{p}\) and the electric field \(\vec{E}\).
Step 3: Detailed Explanation:
For equilibrium, the net torque must be zero. This occurs when \(\tau = pE \sin\theta = 0\), which means \(\sin\theta = 0\). This condition is met for two angles: \(\theta = 0\) and \(\theta = \pi\).
To determine stability, we look at the potential energy \(U = -pE \cos\theta\).
- Stable equilibrium occurs when the potential energy is at a minimum. \(U\) is minimum when \(\cos\theta\) is maximum (\(\cos\theta = +1\)). This happens when \(\theta = 0\).
- Unstable equilibrium occurs when the potential energy is at a maximum. \(U\) is maximum when \(\cos\theta\) is minimum (\(\cos\theta = -1\)). This happens when \(\theta = \pi\).
Step 4: Final Answer:
The dipole is in stable equilibrium when the potential energy is minimum, which occurs when the angle between \(\vec{p}\) and \(\vec{E}\) is 0. This means the dipole moment is aligned with the electric field. Option (D) is correct.