An electric dipole of dipole moment `vec(P)` is placed in a uniform electric field `vec(E)` such that `vecP` is perpendicular to `vecE` The work done to. turn the dipole through an angle of `180^@` is

Options

A. zero
B. pE
C. 2pE
D. `sqrt(2) pE`

Answer

To solve the problem of calculating the work done to turn an electric dipole through an angle of \(180^\circ\) when it is initially perpendicular to a uniform electric field, we can follow these steps:

### Step 1: Understand the Initial Configuration
- The electric dipole moment \(\vec{P}\) is initially perpendicular to the electric field \(\vec{E}\). This means that the angle \(\theta\) between \(\vec{P}\) and \(\vec{E}\) is \(90^\circ\).

### Step 2: Write the Expression for Potential Energy
- The potential energy \(U\) of an electric dipole in a uniform electric field is given by the formula:
\[ U = -\vec{P} \cdot \vec{E} = -PE \cos \theta \]
where \(P\) is the magnitude of the dipole moment, \(E\) is the magnitude of the electric field, and \(\theta\) is the angle between \(\vec{P}\) and \(\vec{E}\).

### Step 3: Calculate Initial Potential Energy
- For the initial configuration where \(\theta = 90^\circ\):
\[ U_i = -PE \cos(90^\circ) = -PE \cdot 0 = 0 \]

### Step 4: Determine the Final Configuration
- When the dipole is turned through \(180^\circ\), the new angle \(\theta\) becomes \(90^\circ + 180^\circ = 270^\circ\).

### Step 5: Calculate Final Potential Energy
- For the final configuration where \(\theta = 270^\circ\):
\[ U_f = -PE \cos(270^\circ) = -PE \cdot 0 = 0 \]

### Step 6: Calculate the Work Done
- The work done \(W\) in turning the dipole is equal to the change in potential energy:
\[ W = U_f - U_i = 0 - 0 = 0 \]

### Conclusion
- The work done to turn the dipole through an angle of \(180^\circ\) is \(0\).