Question

Three electromagnetic waves travel through a certain point P along an x-axis. They are polarized parallel to a y-axis, with the following variations in their amplitudes. Find their resultant at P.

$$\begin{gathered} {\mathit{E}}_{\mathbf{1}}\mathbf{=}\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu V}\mathbf{/}\mathbf{m}\mathbf{)}\mathbf{sin}\left[\left(2\times {10}^{14}\right)t\right] \\ {\mathit{E}}_{\mathbf{2}}\mathbf{=}\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu V}\mathbf{/}\mathbf{m}\mathbf{)}\mathbf{sin}\left[\left(2\times {10}^{14}\right)t+45°\right] \\ {\mathit{E}}_{\mathbf{3}}\mathbf{=}\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu V}\mathbf{/}\mathbf{m}\mathbf{)}\mathbf{sin}\left[\left(2\times {10}^{14}\right)t-45°\right] \end{gathered}$$

Short answer

The resultant amplitude of all waves at point P is$17.1\mathrm{\mu V}/\mathrm{m}$

Step-by-step solution

Identification of given data

The electric field amplitude of the first wave is $A_1=10\mathrm{\mu V}/\mathrm{m}$.

The electric field amplitude of the second wave is role="math" localid="1663147301551" $A_2=5\mathrm{\mu V}/\mathrm{m}$.

The electric field amplitude of the third wave is role="math" localid="1663147316555" $A_3=5\mathrm{\mu V}/\mathrm{m}$.

The phase angle between first and second wave is ${\varphi }_1=45°$.

The phase angle between second and third wave is ${\varphi }_2=90°$.

The phase angle between first and third wave is ${\varphi }_3=45°$.

Understanding the concept

The amplitude of the resultant wave is equal to the vector sum of the amplitude of each wave.

Determination of resultant amplitude of all waves at point P

The resultant amplitude of all wavesat point P is given as:

$E_r=\sqrt{A_1^2+A_2^2+A_3^2+2A_1{A}_2\cos {\varphi }_1+2A_2{A}_3\cos {\varphi }_2+2A_3{A}_1\cos {\varphi }_3}$

Substitute all the values in equation.

Hence, the resultant amplitude of all waves at point P is $17.1\mathrm{\mu V}/\mathrm{m}$.