Question

A dipole is kept in an electric field \( \vec{E}_1 = E_0 \hat{i} \). It oscillates with frequency \( f_1 \). A new electric field \( \vec{E}_2 = 2E_0 \hat{j} + 2E_0 \hat{k} \) is superimposed. The dipole now oscillates in the direction of \( \vec{E}_{\text{net}} \) with frequency \( f_2 \). Find the percentage change in frequency.

• A. 100%
• B. 200%
• C. 50%
• D. 73%

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The frequency of oscillation for a dipole in a uniform electric field is proportional to the square root of the electric field magnitude (\( f \propto \sqrt{E} \)).

Step 2: Key Formula or Approach:
\[ f = \frac{1}{2\pi} \sqrt{\frac{pE}{I}} \implies f \propto \sqrt{E} \]

Step 3: Detailed Explanation:
1. Initial field magnitude: \( E_1 = E_0 \). 2. Net electric field: \( \vec{E}_{net} = E_0 \hat{i} + 2E_0 \hat{j} + 2E_0 \hat{k} \). 3. Magnitude of \( E_{net} \): \[ E_{net} = \sqrt{E_0^2 + (2E_0)^2 + (2E_0)^2} = \sqrt{E_0^2 + 4E_0^2 + 4E_0^2} = \sqrt{9E_0^2} = 3E_0 \] 4. Ratio of frequencies: \[ \frac{f_2}{f_1} = \sqrt{\frac{E_{net}}{E_1}} = \sqrt{\frac{3E_0}{E_0}} = \sqrt{3} \approx 1.732 \] 5. Percentage change: \[ \% \text{change} = \left( \frac{f_2 - f_1}{f_1} \right) \times 100 = (\sqrt{3} - 1) \times 100 \approx 73.2\% \]

Step 4: Final Answer:
The percentage change in frequency is approximately 73%.