Question

Two plane polarized light waves combine at a certain point whose electric field components are $ E_1 = E_0 \sin(\omega t) $ $ E_2 = E_0 \sin(\omega t + \frac{\pi}{3}) $ Find the amplitude of the resultant wave.

• A. \( 0.9 E \)
• B. \( E_0 \)
• C. \( 1.7 E_0 \)
• D. \( 3.4 E_0 \)

Hint:

To find the resultant amplitude of two waves, use the formula for the resultant of two waves with a phase difference. Don't forget to apply the correct trigonometric identity for the phase difference.

Answer 1

The Correct Option is C

We are given two electric field components of plane polarized light: \[ E_1 = E_0 \sin(\omega t) \] \[ E_2 = E_0 \sin\left( \omega t + \frac{\pi}{3} \right) \] The amplitude of the resultant wave is given by the formula: \[ E_{\text{res}} = \sqrt{E_1^2 + E_2^2 + 2E_1 E_2 \cos(\phi)} \] where \( \phi = \frac{\pi}{3} \) is the phase difference between the two waves. Substitute the values: \[ E_{\text{res}} = \sqrt{E_0^2 + E_0^2 + 2E_0^2 \cos\left( \frac{\pi}{3} \right)} \] Since \( \cos\left( \frac{\pi}{3} \right) = \frac{1}{2} \), we get: \[ E_{\text{res}} = \sqrt{E_0^2 + E_0^2 + E_0^2} = \sqrt{3E_0^2} = \sqrt{3} E_0 \]
Thus, the amplitude of the resultant wave is \( \sqrt{3} E_0 \approx 1.7 E_0 \).