Question

A rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec{E}_1 = E_0 \hat{x}$. If another electric field $\vec{E}_2 = 2E_0 (\hat{y} + \hat{z})$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?

• A. 73%
• B. 63%
• C. 83%
• D. 53%

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When a dipole undergoes small angular oscillations in a uniform electric field, the restoring torque relates directly to the strength of the net electric field. Because frequency $\nu$ is proportional to the square root of the electric field magnitude, modifying the net field scales the frequency.

Step 2: Key Formula or Approach:
Restoring torque $\tau = -pE \sin\theta \approx -pE\theta$ (for small angles).
Frequency $f = \frac{1}{2\pi} \sqrt{\frac{pE}{I}}$, meaning $f \propto \sqrt{E_{net}}$.
Percentage change $= \frac{f_{final} - f_{initial}}{f_{initial}} \times 100$.

Step 3: Detailed Explanation:
The initial electric field is $\vec{E}_1 = E_0 \hat{x}$.
Initial magnitude $E_{initial} = \sqrt{E_0^2} = E_0$.
The initial frequency is $f_1 \propto \sqrt{E_0}$.
After adding the second electric field $\vec{E}_2 = 2E_0 \hat{y} + 2E_0 \hat{z}$, the new net electric field is the vector sum:
$\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 = E_0 \hat{x} + 2E_0 \hat{y} + 2E_0 \hat{z}$.
Calculate the magnitude of the new net electric field:
$E_{final} = |\vec{E}_{net}| = \sqrt{E_0^2 + (2E_0)^2 + (2E_0)^2}$
$E_{final} = \sqrt{E_0^2 + 4E_0^2 + 4E_0^2} = \sqrt{9E_0^2} = 3E_0$.
The final frequency is $f_2 \propto \sqrt{E_{final}} = \sqrt{3E_0}$.
Taking the ratio of the frequencies:
$f_2 = \sqrt{3} f_1$.
Calculate the percentage change in frequency:
$% \text{ change} = \left( \frac{f_2 - f_1}{f_1} \right) \times 100%$
$= \left( \frac{\sqrt{3} f_1 - f_1}{f_1} \right) \times 100%$
$= (\sqrt{3} - 1) \times 100%$
Using the approximation $\sqrt{3} \approx 1.732$:
$% \text{ change} = (1.732 - 1) \times 100% = 0.732 \times 100% = 73.2%$.
The approximate percentage change is 73%.

Step 4: Final Answer:
The percentage change is roughly 73%.