An electric dipole of moment $P$ is placed along the direction of electric field $E$. The work done in deflecting the dipole through $180^\circ$ is equal to

Question

For an electric dipole in a non-uniform electric field with dipole moment parallel to direction of the field, the force $F$ and torque $\tau$ on the dipole respectively are __________. Fill in the blank with the correct answer from the options given below.

Answer 1

Step 1: Understanding the Concept:
An electric dipole consists of two equal and opposite charges separated by a small distance.
When placed in an external electric field ($E$), the field exerts a torque on the dipole that tries to align its dipole moment ($P$) with the field.
Work must be done by an external agent to rotate the dipole against this electric torque.
The work done is equal to the change in potential energy ($U$) of the dipole.
Step 2: Key Formula or Approach:
The potential energy of a dipole at an angle $\theta$ with the electric field is:
\[ U = -\vec{P} \cdot \vec{E} = -PE \cos\theta \]
The work done ($W$) in rotating from $\theta_1$ to $\theta_2$ is:
\[ W = U_2 - U_1 = PE (\cos\theta_1 - \cos\theta_2) \]
Step 3: Detailed Explanation:
Initial configuration: The dipole is "placed along the direction of the field". This implies $\theta_1 = 0^\circ$.
Initial Potential Energy: $U_1 = -PE \cos 0^\circ = -PE(1) = -PE$.
Final configuration: It is deflected "through $180^\circ$", meaning the final angle $\theta_2 = 180^\circ$.
Final Potential Energy: $U_2 = -PE \cos 180^\circ = -PE(-1) = +PE$.
Calculating the Work Done:
\[ W = U_2 - U_1 = (+PE) - (-PE) \]
\[ W = PE + PE = +2PE. \]
Since the potential energy of the system has increased (from a minimum at stable equilibrium to a maximum at unstable equilibrium), the work done by the external agent is positive.
Step 4: Final Answer:
The work done in deflecting the dipole through 180 degrees is +2PE.

List-I	List-II
A. The current lags behind the e.m.f. by \( \frac{\pi}{4} \) in phase	I. A.C. circuit containing only an inductor
B. The e.m.f. lags behind the current by \( \frac{\pi}{4} \) in phase	II. A.C. circuit with resistance and inductance in series
C. The current lags behind the e.m.f. in phase by \( \frac{\pi}{2} \)	III. A.C. circuit containing only a capacitor
D. The e.m.f. lags behind the current in phase by \( \frac{\pi}{2} \)	IV. A.C. circuit with resistance and capacitor in series

An electric dipole of moment \(P\) is placed along the direction of electric field \(E\). The work done in deflecting the dipole through \(180^\circ\) is equal to

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The Correct Option is B

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