Problem electromagnetic waves

Finally, we can write this in terms of real and imaginary parts:\vec{E}=(\vec{E}_{01}+\vec{E}_{02})e^{i(\vec{k}\cdot\vec{r}-\omega t)}(2\cos(\frac{\alpha_1-\alpha_2}{2})\cos(\frac{\alpha_1+\alpha_2}{2})+i2\cos(\frac{\alpha_1-\alpha_2}{2})\sin(\frac{\alpha_1+\alpha_2}{2}))We can see that the real and imaginary parts of the resultant wave are both proportional
  • #1
Petar Mali
290
0

Homework Statement



Two monocromatic waves is given by

[tex]\vec{E_1}=\vec{E}_{01}cos(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)[/tex] and

[tex]\vec{E_2}=\vec{E}_{02}cos(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)[/tex]

are linearly polarized along two normal directions. Taking that waves have equal amplitude, find polarisation of resultant of this two waves.



Homework Equations





The Attempt at a Solution



I suppose that I must try with

[tex]\vec{E_1}=Re\{\vec{E}_{01}e^{i(\vec{k}\cdot \vec{r}-\omega t+\alpha_1)}\}[/tex]

[tex]\vec{E_2}=Re\{\vec{E}_{02}e^{i(\vec{k}\cdot \vec{r}-\omega t+\alpha_2)}\}[/tex]

[tex]\vec{E}=Re\{(\vec{E}_{01}+\vec{E}_{02}e^{i\xi})e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)}\}[/tex]

where [tex]\xi=\alpha_2-\alpha_1[/tex]. What now?
 
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  • #2




Thank you for your forum post. Your approach seems to be on the right track. Let's continue with your attempt at a solution and see if we can find the polarization of the resultant wave. First, let's rewrite the equations using complex numbers for simplicity:

\vec{E_1}=\vec{E}_{01}e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)}

\vec{E_2}=\vec{E}_{02}e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_2)}

\vec{E}=(\vec{E}_{01}+\vec{E}_{02}e^{i\xi})e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)}

Next, let's combine the two equations for \vec{E_1} and \vec{E_2} to find the resultant wave \vec{E}:

\vec{E}=\vec{E}_{01}e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_1)}+\vec{E}_{02}e^{i(\vec{k}\cdot\vec{r}-\omega t+\alpha_2)}

Since we are assuming equal amplitudes for both waves, we can factor out \vec{E}_{01} and \vec{E}_{02} from the equation:

\vec{E}=(\vec{E}_{01}+\vec{E}_{02})e^{i(\vec{k}\cdot\vec{r}-\omega t)}(e^{i\alpha_1}+e^{i\alpha_2})

Now, we can use Euler's formula to rewrite the last term as:

e^{i\alpha_1}+e^{i\alpha_2}=2\cos(\frac{\alpha_1-\alpha_2}{2})e^{i\frac{\alpha_1+\alpha_2}{2}}

Substituting this back into our equation for \vec{E}, we get:

\vec{E}=(\vec{E}_{01}+\vec{E}_{02})e^{i(\vec{k}\cdot\vec{r}-\omega t)}(2\cos(\frac{\alpha_1-\alpha_2}{2})
 

1. What are electromagnetic waves?

Electromagnetic waves are a type of energy that can travel through space. They are produced by the acceleration of electric charges and consist of electric and magnetic fields oscillating at right angles to each other.

2. What are the sources of problem electromagnetic waves?

The most common sources of problem electromagnetic waves include power lines, electronic devices, and radio and TV antennas. Other sources can include electric motors, transformers, and even natural phenomena such as lightning.

3. How do electromagnetic waves cause problems?

Electromagnetic waves can cause problems by interfering with other electronic devices, disrupting communication signals, and causing health concerns. They can also affect the performance of sensitive equipment and cause power outages.

4. How can we protect ourselves from problem electromagnetic waves?

To protect ourselves from problem electromagnetic waves, we can use shielding materials, such as metal or conductive fabrics, to block or redirect the waves. We can also limit our exposure by keeping a safe distance from sources and using devices with lower electromagnetic emissions.

5. What research is being done to address problem electromagnetic waves?

Scientists are conducting research to better understand the effects of electromagnetic waves and develop technologies to mitigate their negative impacts. This includes studying the health effects of exposure, finding ways to reduce emissions, and developing better shielding materials.

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