The frequency of oscillation of an electric dipole moment having dipole moment p and rotational inertia l, oscillating in a uniform electric field E is given

Options

A. `(1//2pi) sqrt(I//pE)`
B. `(1//2pi)sqrt(pE//I)`
C. `(2pi)sqrt(pE//I)`
D. `(2pi)sqrt(I//pE)`

Answer

To find the frequency of oscillation of an electric dipole in a uniform electric field, we can follow these steps:

### Step 1: Understand the System
An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment \( p \) is defined as:
\[ p = q \cdot d \]
where \( q \) is the charge and \( d \) is the separation between the charges.

### Step 2: Apply Torque on the Dipole
When the dipole is placed in a uniform electric field \( E \) and is tilted at an angle \( \theta \), a torque \( \tau \) acts on it given by:
\[ \tau = p \cdot E \cdot \sin(\theta) \]

### Step 3: Relate Torque to Angular Acceleration
The torque also relates to the angular acceleration \( \alpha \) through the moment of inertia \( I \):
\[ \tau = I \cdot \alpha \]
Thus, we can write:
\[ p \cdot E \cdot \sin(\theta) = I \cdot \alpha \]

### Step 4: Small Angle Approximation
For small angles \( \theta \), we can use the approximation:
\[ \sin(\theta) \approx \theta \]
This simplifies our equation to:
\[ p \cdot E \cdot \theta = I \cdot \alpha \]

### Step 5: Express Angular Acceleration
The angular acceleration \( \alpha \) can be expressed as:
\[ \alpha = \frac{d^2\theta}{dt^2} \]
Substituting this into our equation gives:
\[ p \cdot E \cdot \theta = I \cdot \frac{d^2\theta}{dt^2} \]

### Step 6: Rearranging the Equation
Rearranging the equation leads to:
\[ \frac{d^2\theta}{dt^2} + \frac{p \cdot E}{I} \theta = 0 \]
This is a standard form of the simple harmonic motion equation.

### Step 7: Identify the Angular Frequency
From the standard form, we can identify the angular frequency \( \omega \):
\[ \omega^2 = \frac{p \cdot E}{I} \]
Thus, the angular frequency \( \omega \) is:
\[ \omega = \sqrt{\frac{p \cdot E}{I}} \]

### Step 8: Relate Angular Frequency to Frequency
The frequency \( f \) of oscillation is related to the angular frequency by:
\[ \omega = 2\pi f \]
Therefore, we can express the frequency as:
\[ f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{p \cdot E}{I}} \]

### Final Result
The frequency of oscillation of the electric dipole is given by:
\[ f = \frac{1}{2\pi} \sqrt{\frac{p \cdot E}{I}} \]