Updated spacing between @ latex expressions

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en/12.1.4.md +9 −9
@@ -6,14 +6,14 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta http-equiv="content-language" content="en">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
− <meta name="description" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
+ <meta name="description" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega (t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega (t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
<meta name="author" content="Aliaksandr Melnichenka">
<meta name="date" content="2023-10" scheme="YYYY-MM">
− <meta property="og:title" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
+ <meta property="og:title" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega (t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega (t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
<meta property="og:image" content="img/logo.png">
− <meta property="og:description" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
+ <meta property="og:description" content="Two sinusoidal waves with the same polarization $E_1~\sin{[\omega (t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega (t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?">
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− <title>Two sinusoidal waves with the same polarization $E_1~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?</title>
+ <title>Two sinusoidal waves with the same polarization $E_1~\sin{[\omega (t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega (t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?</title>
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@@ -49,21 +49,21 @@
<h3> Statement </h3>
<p>
$12.1.4$
− Two sinusoidal waves with the same polarization $E_1~\sin{[\omega(t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega(t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?
+ Two sinusoidal waves with the same polarization $E_1~\sin{[\omega (t-z/c)+\varphi_1]}$, $E_2~\sin{[\omega (t-z/c)+\varphi_2]}$ are superimposed on each other. What is the amplitude of the electric field strength of the resulting wave? What is the phase of this wave?
</p>
<h3>Solution</h3>
<p>
Since waves are superimposed,
$$E_R = E_{w1} + E_{w2}$$
− $$E_R = E_1~\sin{[\omega(t-z/c)+\varphi_1]} + E_2~\sin{[\omega(t-z/c)+\varphi_2]}$$
− $$E_R = (E_1\cos{\varphi_1}+E_2\cos{\varphi_2})\sin{[\omega(t-z/c)]} + (E_1\sin{\varphi_1}+E_2\sin{\varphi_2})\cos{[\omega(t-z/c)]}$$
+ $$E_R = E_1~\sin{[\omega (t-z/c)+\varphi_1]} + E_2~\sin{[\omega (t-z/c)+\varphi_2]}$$
+ $$E_R = (E_1\cos{\varphi_1}+E_2\cos{\varphi_2})\sin{[\omega (t-z/c)]} + (E_1\sin{\varphi_1}+E_2\sin{\varphi_2})\cos{[\omega (t-z/c)]}$$
Let's suppose that
$$E_1\cos{\varphi_1}+E_2\cos{\varphi_2} = E \cos{\varphi} = E_x$$
and
$$E_1\sin{\varphi_1}+E_2\sin{\varphi_2} = E \sin{\varphi} = E_y$$
and considering the trigonometric identity $\sin{(x+y)} = \sin{x}\cos{y} + \cos{x}\sin{y}$,
− $$E_R = E \sin{[\omega(t-z/c)+\varphi]}$$
+ $$E_R = E \sin{[\omega (t-z/c)+\varphi]}$$
As $E = \sqrt{{E_x}^2 + {E_y}^2}$ and taking in account that $\cos{(x-y)} = \cos{x}\cos{y} + \sin{x}\sin{y}$
</p>
<h4>Answer 1</h4>
@@ -77,7 +77,7 @@
</p>
<h4>Answer 2</h4>
<p>
− $$\Phi = \omega(t-z/c) + \arctan {\frac{E_1\sin{\varphi_1}+E_2\sin{\varphi_2}}{E_1\cos{\varphi_1}+E_2\cos{\varphi_2}}}$$
+ $$\Phi = \omega (t-z/c) + \arctan {\frac{E_1\sin{\varphi_1}+E_2\sin{\varphi_2}}{E_1\cos{\varphi_1}+E_2\cos{\varphi_2}}}$$
</p>
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