An electric dipole located in uniform electric field in stable equilibrium position . It is now slowly rotated to the position of unstable equilibrium . Work donw by the external agent in the process is numerically how may times the maximum electric torque experienced by the dipole during the process ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial and Final Positions of the Dipole
- The electric dipole is initially in a stable equilibrium position where the dipole moment \( \mathbf{p} \) is aligned with the electric field \( \mathbf{E} \). This means that the angle \( \theta = 0^\circ \).
- The dipole is then rotated to an unstable equilibrium position where the dipole moment is antiparallel to the electric field, which means \( \theta = 180^\circ \).

### Step 2: Calculate the Initial and Final Potential Energy
- The potential energy \( U \) of an electric dipole in a uniform electric field is given by the formula:
\[ U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos(\theta) \]
- For the stable equilibrium position (\( \theta = 0^\circ \)):
\[ U_{\text{initial}} = -pE \cos(0) = -pE \]
- For the unstable equilibrium position (\( \theta = 180^\circ \)):
\[ U_{\text{final}} = -pE \cos(180^\circ) = -pE \cdot (-1) = pE \]

### Step 3: Calculate the Change in Potential Energy
- The work done by the external agent in rotating the dipole is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = pE - (-pE) = pE + pE = 2pE \]

### Step 4: Calculate the Maximum Torque
- The torque \( \tau \) experienced by the dipole in an electric field is given by:
\[ \tau = \mathbf{p} \times \mathbf{E} = pE \sin(\theta) \]
- The maximum torque occurs when \( \sin(\theta) = 1 \) (i.e., \( \theta = 90^\circ \)):
\[ \tau_{\text{max}} = pE \]

### Step 5: Relate Work Done to Maximum Torque
- We have calculated that the work done \( W = 2pE \) and the maximum torque \( \tau_{\text{max}} = pE \).
- To find how many times the maximum torque the work done is, we can express it as:
\[ \text{Number of times} = \frac{W}{\tau_{\text{max}}} = \frac{2pE}{pE} = 2 \]

### Final Answer
The work done by the external agent in the process is numerically **2 times** the maximum electric torque experienced by the dipole during the process.

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