The work done in deflecting a dipole through `180^(@)` from field direction is

Options

A. perpendicular to each other
B. 2PE
C. `(1)/(2)PE`
D. zero

Explanation

To find the work done in deflecting a dipole through \(180^\circ\) from the direction of the electric field, we can follow these steps:

### Step 1: Understand the Dipole and Electric Field
A dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance \(2L\). The dipole moment \(P\) is given by:
\[ P = Q \cdot 2L \]
The dipole is placed in an electric field \(E\).

### Step 2: Determine the Torque on the Dipole
The torque \(\tau\) acting on the dipole in an electric field is given by:
\[ \tau = P E \sin \theta \]
where \(\theta\) is the angle between the dipole moment and the electric field direction.

### Step 3: Work Done in Rotating the Dipole
The work done \(W\) in rotating the dipole from an angle \(\theta_1\) to \(\theta_2\) is given by the integral of torque over the angle:
\[ W = \int_{\theta_1}^{\theta_2} \tau \, d\theta \]
Substituting the expression for torque:
\[ W = \int_{\theta_1}^{\theta_2} P E \sin \theta \, d\theta \]

### Step 4: Integrate the Torque Expression
The integral of \(\sin \theta\) is \(-\cos \theta\):
\[ W = P E \left[-\cos \theta\right]_{\theta_1}^{\theta_2} = P E \left(-\cos \theta_2 + \cos \theta_1\right) \]

### Step 5: Substitute the Limits for the Integral
If we are rotating the dipole through \(180^\circ\), we can set \(\theta_1 = 0^\circ\) (initially aligned with the electric field) and \(\theta_2 = 180^\circ\):
\[ W = P E \left(-\cos 180^\circ + \cos 0^\circ\right) \]
Knowing that \(\cos 180^\circ = -1\) and \(\cos 0^\circ = 1\):
\[ W = P E \left(-(-1) + 1\right) = P E (1 + 1) = 2 P E \]

### Step 6: Conclusion
Thus, the work done in deflecting the dipole through \(180^\circ\) from the direction of the electric field is:
\[ W = 2 P E \]