An electric dipole of moment `vec(P)` is placed in a uniform electric field `vec(E)` such that `vec(P)` points along `vec(E)` . If the dipole is slightly rotated about an axis perpendicular to the plane containing `vec(E) and vec(P)` and passing through the centre of the dipole, the dipole executes simple harmonic motion. Consider I to be the moment of inertia of the dipole about the axis of rotation. What is the time period of such oscillation ?

Options

A. `sqrt((PE//I))`
B. ` 2 pi sqrt((I//PE))`
C. `2 pi sqrt((I // 2 PE))`
D. None of these

Explanation

To find the time period of the oscillation of an electric dipole in a uniform electric field when it is slightly rotated, we can follow these steps:

### Step 1: Understand the System
An electric dipole with a dipole moment \(\vec{P}\) is placed in a uniform electric field \(\vec{E}\). The dipole moment is aligned with the electric field. When the dipole is slightly rotated by an angle \(\theta\), it experiences a torque due to the electric field.

### Step 2: Calculate the Torque
The torque \(\tau\) acting on the dipole in the electric field is given by:
\[ \tau = \vec{P} \times \vec{E} = P E \sin(\theta) \]
For small angles, we can approximate \(\sin(\theta) \approx \theta\) (in radians). Thus, the torque becomes:
\[ \tau \approx P E \theta \]

### Step 3: Relate Torque to Angular Acceleration
According to Newton's second law for rotation, the torque is also related to the moment of inertia \(I\) and angular acceleration \(\alpha\):
\[ \tau = I \alpha \]
Substituting the expression for torque, we get:
\[ P E \theta = I \alpha \]
Since angular acceleration \(\alpha\) can be expressed as \(\alpha = \frac{d^2\theta}{dt^2}\), we rewrite the equation:
\[ P E \theta = I \frac{d^2\theta}{dt^2} \]

### Step 4: Formulate the Equation of Motion
Rearranging the equation gives us:
\[ \frac{d^2\theta}{dt^2} + \frac{P E}{I} \theta = 0 \]
This is the standard form of a simple harmonic motion equation, where \(\omega^2 = \frac{P E}{I}\).

### Step 5: Find the Angular Frequency
From the equation, we can identify the angular frequency \(\omega\):
\[ \omega = \sqrt{\frac{P E}{I}} \]

### Step 6: Calculate the Time Period
The time period \(T\) of simple harmonic motion is related to the angular frequency by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the expression for \(\omega\):
\[ T = 2\pi \sqrt{\frac{I}{P E}} \]

### Final Result
Thus, the time period of the oscillation of the dipole is:
\[ T = 2\pi \sqrt{\frac{I}{P E}} \]

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