But first, let's go over this problem.

但首先，讓我們來看看這個問題。

Light travels from air to water.

光從空氣傳播到水。

At what incident angle is the reflected light completely polarized?

入射角度為多少時，反射光完全偏振？

Well, let's find out.

讓我們一探究竟。

So let's say we have air on top and water beneath.

假設上面是空氣，下面是水。

And so here we have the normal line.

於是，我們就有了正常的線路。

And this is going to be the incident ray.

這將是事件射線。

And this is the reflected ray.

這就是反射光線。

And that's going to be the refracted ray, which may not be so narrow.

這將是折射光線，可能不會那麼窄。

So this is the incident angle.

這就是入射角。

This is the reflected angle.

這就是反射角。

And this is the refracted angle.

這就是折射角。

Now according to the law of reflection, the incident angle is equal to the reflected angle.

根據反射定律，入射角等於反射角。

And according to Snell's law of refraction, n1 sine theta 1, or the incident angle, is equal to n2 sine theta 2, or the refracted angle.

根據斯涅爾折射定律，n1 正弦θ 1 或入射角等於 n2 正弦θ 2 或折射角。

Let's say this is n1 and this is n2.

假設這是 n1，這是 n2。

Now for most incident angles, the reflected angle is partially polarized.

現在，對於大多數入射角來說，反射角是部分偏振的。

However, there is a specific incident angle where the reflected angle is completely polarized.

不過，有一個特定的入射角，其反射角是完全偏振的。

So our goal is to calculate that angle.

是以，我們的目標就是計算出這個角度。

That angle is known as the polarizing angle.

這個角度被稱為偏振角。

And so that's theta p.

這就是θ p。

Now because the angle of incidence is equal to the angle of reflection, we can call both of these angle p.

現在，由於入射角等於反射角，我們可以把這兩個角度都叫做 p。

Now this is going to stay the way it is.

現在的情況將保持不變。

So how can we calculate this angle?

那麼，如何計算這個角度呢？

There's something that you need to know about the reflected ray and the refracted ray in order to calculate that angle.

要計算這個角度，你需要知道反射光線和折射光線的一些資訊。

And that is that at the polarizing angle, these two rays are at right angles.

也就是說，在偏振角處，這兩條光線成直角。

They're 90 degrees with respect to each other.

它們相互成 90 度。

And a full line is 180 degrees.

而全線是 180 度。

So 180 minus 90 means that these two are complementary.

是以，180 減 90 意味著這兩者是互補的。

They add up to 90.

加起來是 90。

So we can say that theta p plus theta r is equal to 90.

是以我們可以說，θ p 加 θ r 等於 90。

Now even though the reflected ray is completely polarized, it's good to know that the refracted ray is only partially polarized.

現在，儘管反射光線是完全偏振的，但很高興知道折射光線只是部分偏振。

So let's get back to this formula.

讓我們回到這個公式上來。

So theta p is 90 minus theta r.

是以，θ p 是 90 減去θ r。

So the polarizing angle is 90 degrees minus the refracted angle.

是以，偏振角是折射角減去 90 度。

Now let's go back to Snell's Law.

現在讓我們回到斯內爾定律。

So n1 sine theta p, the incident angle is the same as the polarizing angle.

是以 n1 正弦 p，入射角與偏振角相同。

And that's equal to n2 times sine theta r.

這等於 n2 乘以正弦θ r。

Well, you know what, I should have solved for theta r instead.

好吧，你知道嗎，我本該求解 theta r 的。

So let's do that.

那就這麼辦吧。

So theta r is going to be 90 minus theta p.

是以，θ r 將是 90 減去θ p。

So this is what I needed.

這就是我所需要的。

So now going back to this equation, it's going to be n1 sine theta p is equal to n2.

是以，現在回到這個等式，它將是 n1 正弦 theta p 等於 n2。

And instead of sine theta r, we can replace that with 90 minus theta p.

我們可以用 90 減θ p 來代替正弦θ r。

Now what do we need to do with that?

現在我們該怎麼做呢？

There's something that we need to do, but what is it exactly that we can do with this?

我們需要做一些事情，但我們到底能做什麼呢？

Perhaps you have taken trigonometry, and if you have, there's something called co-function identities.

也許你學過三角學，如果你學過，就會知道有一種叫做共函數等式的東西。

So cosine theta is equal to sine 90 minus theta.

是以，餘弦θ 等於正弦 90 減去θ。

So cosine is the co-function of sine.

是以，餘弦是正弦的共函數。

So whenever the two angles add up to 90, sine and cosine are equal.

是以，只要兩個角相加等於 90，正弦和餘弦就相等。

So for example, cosine of 10 degrees is equal to sine of 80, because 10 plus 80 is 90.

例如，10 度的餘弦等於 80 度的正弦，因為 10 加 80 等於 90。

Cosine 20 is equal to sine of 70.

餘弦 20 等於正弦 70。

And you can confirm this with your calculator.

您可以用計算器確認這一點。

And cosine 30 degrees is equal to sine 90 minus 30, or sine 60.

餘弦 30 度等於正弦 90 減 30，即正弦 60。

And so we can replace sine 90 minus theta p with cosine p.

是以，我們可以用餘弦 p 代替正弦 90 減 Theta p。

So now we have this equation.

現在我們有了這個等式。

So at this point, what I'm going to do is I'm going to divide both sides by cosine theta p.

是以，此時我要做的是將兩邊除以餘弦θ p。

So on the right side, these two will cancel.

是以，在右側，這兩者將相互抵消。

On the left side, sine divided by cosine is tangent.

在左邊，正弦除以餘弦就是正切。

And so if we divide both sides by n1, now we can get the polarizing angle.

是以，如果我們把兩邊都除以 n1，就能得到偏振角。

So tangent theta p is equal to n2 over n1.

是以，切向量 p 等於 n2 大於 n1。

Now you need to know the direction in which light travels.

現在你需要知道光的傳播方向。

So light is going to travel from n1 to a material with an index of refraction of n2.

是以，光會從 n1 傳播到折射率為 n2 的材料。

So as we saw, it went from air to water.

是以，正如我們所看到的，它從空氣變成了水。

So n1 is going to be air, because that's where the light is coming from.

是以，n1 將是空氣，因為光是從那裡發出的。

And it's going to water as it refracts to it.

當它折射到水面時，就會變成水。

So n2 is for water.

所以 n2 代表水。

Now n for water is actually 1.33.

現在水的 n 實際上是 1.33。

When I wrote 2, I was meaning like n2.

當我寫 2 時，我的意思是 n2。

So let's fix that.

所以，讓我們來解決這個問題。

Here it is for water.

這裡是水。

So now let's calculate the incident angle at which the reflected light is completely polarized.

現在我們來計算反射光完全偏振的入射角度。

So we need to calculate theta p.

是以，我們需要計算 Theta p。

So therefore, we need to use the arctangent function.

是以，我們需要使用反正切函數。

So the polarizing angle is going to be arctan n2 divided by n1.

是以，偏振角將是 arctan n2 除以 n1。

By the way, this relation is known as Brewster's law.

順便提一下，這種關係被稱為布魯斯特定律。

And sometimes this is referred to as Brewster's angle, particularly when air is involved.

有時這也被稱為布魯斯特角，尤其是在涉及空氣的情況下。

So if you need to calculate Brewster's angle, you could simply use that formula.

是以，如果您需要計算布儒斯特角，只需使用該公式即可。

It's the equivalent of the polarizing angle.

這相當於偏振角。

So in this case, we're going from air to water.

是以，在這種情況下，我們要從空氣轉向水。

So n2 is associated with water, so that's going to be 1.33.

是以，n2 與水有關，所以會是 1.33。

And n1, that's for air, so that's 1.

n1 代表空氣，所以是 1。

So it's arctangent, 1.33.

所以是反正切，1.33。

So theta p is 53.06 degrees.

是以，θ p 是 53.06 度。

Now what is the angle of refraction for which light is transmitted to the water?

現在，光線透射到水中的折射角是多少？

So we can use Snell's law to get the answer if we want to.

是以，如果我們願意，可以使用斯涅耳定律來得到答案。

So this is the polarizing angle.

這就是偏振角。

So n1 is 1 times sine of 53.06, and n2 is 1.33 times sine theta r.

是以，n1 是正弦 53.06 的 1 倍，n2 是正弦 rta 的 1.33 倍。

I'm going to show you another way to get the answer, but I want you to be familiar with both ways.

我將向你展示另一種獲取答案的方法，但我希望你熟悉這兩種方法。

The second way is easier, by the way, so just keep that in mind.

順便說一句，第二種方法更簡單，請記住這一點。

So let's divide both sides by 1.33.

是以，我們把兩邊都除以 1.33。

So it's going to be sine 53.06 divided by 1.33, and so that's 0.601, and that's equal to sine theta r.

是以，正弦 53.06 除以 1.33 就是 0.601，等於正弦 theta r。

Now theta r, the refracted angle, that's going to be arcsine of 0.601.

現在，θ r，即折射角，將是 0.601 的餘弦值。

And so that angle is 36.94 degrees.

是以，這個角度是 36.94 度。

Now an easier way to get that answer is to use this formula.

現在，一個更簡單的方法是使用這個公式來得到答案。

Now recall that we said that the refracted angle and the polarizing angle are complementary.

現在回憶一下，我們說過折射角和偏振角是互補的。

They add up to 90.

加起來是 90。

So the refracted angle is just 90 minus the polarizing angle.

是以，折射角就是偏振角減去 90。

So 90 minus 53.06, that will give us the same answer of 36.94 degrees.

所以 90 減去 53.06，得到的答案還是 36.94 度。

So as you can see, that's a more simpler approach to get this angle.

是以，正如你所看到的，這是獲得這個角度的一個更簡單的方法。

Now let's move on to the next problem.

現在，讓我們進入下一個問題。

Number two, what is Brewster's angle for light that travels from air to glass?

第二，從空氣到玻璃的光的布儒斯特角是多少？

So feel free to pause the video if you want to work on this problem.

是以，如果您想解決這個問題，可以暫停視頻。

So we're going from air to glass.

所以，我們要從空氣變成玻璃。

And the index of refraction of air, that's going to be 1.

空氣的折射率為 1。

Glass is going to be N2 because light is traveling from air to glass.

玻璃會變成 N2，因為光會從空氣傳播到玻璃。

And so the index of refraction for glass is 1.5.

是以，玻璃的折射率為 1.5。

So here we have the incident angle, the reflected angle, and the refracted angle.

是以，我們有了入射角、反射角和折射角。

So both of these will be considered to be theta p, and this is theta r.

是以，這兩個都將被視為 theta p，而這就是 theta r。

So theta p is the same as the Brewster's angle, so you can replace theta p with theta b if you want to.

是以，θ p 與布儒斯特角相同，所以如果你願意，可以用θ b 代替θ p。

So the Brewster's angle is equal to N2 over N1.

是以，布儒斯特角等於 N2 大於 N1。

Well that's tangent of the Brewster angle.

這就是布魯斯特角度的切入點。

The Brewster angle itself is arc tangent, N2 over N1.

布儒斯特角本身就是弧切，即 N2 大於 N1。

So N2 in this example is 1.5, and 1 is 1.

是以，本例中的 N2 是 1.5，而 1 是 1。

So it's just the arc tangent of 1.5.

所以它只是 1.5 的弧切值。

And so that's going to be 56.3 degrees.

是以，這將是 56.3 度。

Now if you need to calculate the angle of refraction, it's going to be 90 degrees minus the Brewster angle, or the polarizing angle.

現在，如果你需要計算折射角，它將是 90 度減去布儒斯特角，或者偏振角。

And so that's going to be 33.7 degrees.

是以，這將是 33.7 度。

But this is the answer that we're looking for in this problem.

但這就是我們在這個問題中尋找的答案。

So that's how you can calculate the Brewster's angle for a simple problem like this.

這就是如何計算布儒斯特角的簡單方法。

Number three, what is the polarizing angle for light that travels from diamond to glass?

第三，從鑽石到玻璃的光的偏振角是多少？

So we're going to say this is diamond, and here we have glass.

是以，我們可以說這是鑽石，這裡是玻璃。

So this is the incident ray, here we have the reflected ray, and a refracted ray.

這就是入射光線、反射光線和折射光線。

Actually this should be wider, because we're going from a high index of refraction to a material with a low index of refraction.

實際上，這個範圍應該更大，因為我們要從高折射率的材料轉換到低折射率的材料。

So the ray is going to bend away from the normal line.

是以，射線會偏離法線彎曲。

So this is going to be theta p and theta r.

是以，這將是 theta p 和 theta r。

So N1 is for diamond, we're starting from that side first, so that's going to be 2.42.

是以，N1 代表鑽石，我們先從這一邊開始，所以是 2.42。

And N2 is glass, which is 1.5.

而 N2 是玻璃，是 1.5。

So the polarizing angle, tangent of the polarizing angle, is N2 over N1.

是以，偏振角（偏振角的正切值）是 N2 大於 N1。

And the polarizing angle is going to be arc tangent of N2 divided by N1.

偏振角將是 N2 的弧形正切除以 N1。

So in this case, N2 is 1.5, N1 is 2.42.

是以，在這種情況下，N2 是 1.5，N1 是 2.42。

And so the polarizing angle is 31.8 degrees.

是以偏振角為 31.8 度。

The refracted angle is going to be 90 degrees minus the polarizing angle.

折射角將是偏振角減去 90 度。

So that's 90 minus 31.8, and so that's going to be 58.2 degrees.

是以，90 減去 31.8，就是 58.2 度。

So now you know how to calculate the polarizing angle and the refracted angle.

現在你知道如何計算偏振角和折射角了吧。

And so that's it for this video.

本視頻到此結束。

Thanks for watching, and have a good day.

感謝您的收看，祝您愉快。

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