Question

Two plane polarized light waves combine at a certain point, whose "E" components are: \[ E_1 = E_0 \sin \omega t, \quad E_2 = E_0 \sin \left( \omega t + \frac{\pi}{3} \right) \] Find the amplitude of the resultant wave.

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

Hint:

When two sinusoidal waves combine, the amplitude of the resultant wave is calculated using the formula \( E_R = \sqrt{E_1^2 + E_2^2 + 2 E_1 E_2 \cos(\phi)} \), where \( \phi \) is the phase difference between the waves.

Answer 1

The Correct Option is C

The two electric fields, \( E_1 \) and \( E_2 \), are defined as: \[ E_1 = E_0 \sin \omega t, \quad E_2 = E_0 \sin \left( \omega t + \frac{\pi}{3} \right) \] The resultant amplitude, \( E_R \), of these two waves with identical frequencies is determined by the formula: \[ E_R = \sqrt{E_1^2 + E_2^2 + 2 E_1 E_2 \cos(\phi)} \] In this context, \( \phi \), the phase difference between the waves, is \( \frac{\pi}{3} \). Substituting the given values: \[ E_R = \sqrt{E_0^2 + E_0^2 + 2 E_0 \cdot E_0 \cdot \cos \left( \frac{\pi}{3} \right)} \] Given that \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \): \[ E_R = \sqrt{E_0^2 + E_0^2 + 2 E_0^2 \cdot \frac{1}{2}} = \sqrt{E_0^2 + E_0^2 + E_0^2} = \sqrt{3 E_0^2} \] Consequently, the amplitude of the resultant wave is: \[ E_R = \sqrt{3} E_0 \approx 1.7 E_0 \] The determined amplitude for the resultant wave is \( 1.7 E_0 \).