In this case, we're going to have refraction across two boundaries.

在這種情況下，我們要跨越兩條邊界進行折射。

Here is a block of glass.

這是一塊玻璃。

Index of refraction is 1.5.

折射率為 1.5。

We have our first boundary, we have our second boundary.

我們有第一道邊界，也有第二道邊界。

Here's the incoming ray.

這是射入的射線。

It goes through the glass and then out the glass on the other side.

它穿過玻璃，然後從玻璃的另一面流出。

Notice that I used the numerics that the angle of incidence is theta sub 1, The angle of refraction across the first boundary is theta sub 2, then the angle of incidence on the second boundary is theta sub 3, and the angle of refraction across the second boundary is theta sub 4.

請注意，我使用的數字表示入射角為 theta sub 1，穿過第一條邊界的折射角為 theta sub 2，然後第二條邊界的入射角為 theta sub 3，穿過第二條邊界的折射角為 theta sub 4。

In line with that, the index of refraction on this side of the first boundary is n sub 1, the index of refraction on the other side of that boundary is n sub 2, the index of refraction on this side of the second boundary is n sub 3, which by the way of course is the same as n sub 2, as you're still inside the glass.

據此，第一條邊界這邊的折射率是 n sub 1，邊界另一邊的折射率是 n sub 2，第二條邊界這邊的折射率是 n sub 3，順便說一下，這當然與 n sub 2 相同，因為你還在玻璃裡面。

And then the index of refraction on the second side of the second boundary is n sub 4, which is back to what we started with, with the air.

然後，第二個邊界第二側的折射率是 n sub 4，這又回到了我們開始時的空氣折射率。

Notice that when we have the first boundary, we're traveling from an index of refraction, which is low, to an index of refraction, which is high, which means that the beam bends or refracts towards the normal.

請注意，當我們有第一個邊界時，我們正從折射率低的地方向折射率高的地方移動，這意味著光束向法線彎曲或折射。

Here we're traveling from an index of refraction, which is high, to an index of refraction, which is slow, which means that the beam will refract or bend away from the normal.

在這裡，我們正從折射率高的地方向折射率低的地方移動，這意味著光束會折射或彎曲，偏離法線。

And so the question is, what will theta sub 4 be equal to?

那麼問題來了，θ sub 4 等於多少呢？

To figure that out, we have to start with theta sub 1.

要弄清楚這一點，我們必須從 Theta sub 1 開始。

Now theta sub 1 was not given, you're only given this angle right here.

現在沒有給出 Theta sub 1，只給出了這個角度。

And to help you out, we're going to continue drawing this line horizontally this way.

為了幫助你，我們將繼續這樣水準地畫這條線。

Now notice that this line right here is parallel to this line right there.

現在請注意，這裡的這條線與那邊的這條線是平行的。

And this angle here and this angle here are what we call alternate interior angles, which means that if this angle is 45 degrees, then this angle must be 45 degrees as well.

這裡的這個角和這裡的這個角是我們所說的交替內角，也就是說，如果這個角是 45 度，那麼這個角也一定是 45 度。

And since this here is a right angle by definition, because this is the normal to this side right here, if this is 45 degrees, then this must be 45 degrees as well, which means theta sub 1 is 45 degrees.

根據定義，這裡是一個直角，因為這裡是這邊的法線，如果這裡是 45 度，那麼這裡也一定是 45 度，也就是說，θ sub 1 是 45 度。

And now we can go ahead and try to figure out what theta sub 2 is equal to.

現在，我們可以繼續嘗試找出 Theta sub 2 等於多少。

And for that, we use Snell's Law, n1 sine of theta 1 is equal to n2 sine of theta sub 2.

為此，我們使用斯涅爾定律，即 n1 正弦的θ 1 等於 n2 正弦的θ sub 2。

Since we're looking for theta sub 2, we're going to flip the equation around.

既然我們要找的是 Theta sub 2，我們就要把等式翻轉過來。

So n2 sine of theta 2 equals n1 sine of theta 1.

是以，θ 2 的正弦值 n2 等於θ 1 的正弦值 n1。

Divide both sides by n sub 2, so we get the sine of theta sub 2 is equal to n1 over n2 times the sine of theta 1.

將兩邊都除以 n 次 2，我們就得到了θ 次 2 的正弦等於 n1 乘以 n2 再乘以θ 1 的正弦。

And finally, theta sub 2, therefore, is equal to the arc sine or the inverse sine of n1 over n2 times the sine of theta sub 1.

最後，θ sub 2 等於 n1 乘以 n2 的弧正弦或 n1 乘以 n2 的逆正弦乘以θ sub 1 的正弦。

And we plug in the numbers, it's the arc sine of n1 is equal to 1 and 2 is equal to 1.5.

輸入數字後，n1 的弧正弦等於 1，2 等於 1.5。

And notice how important it is that you do all the notations correctly right from the start.

請注意，從一開始就正確地完成所有記號是多麼重要。

It makes it a lot easier to plug the numbers into the equation.

這樣就能更容易地將數字輸入等式。

And of course, sine of theta 1 is the sine of 45 degrees.

當然，θ 1 的正弦就是 45 度的正弦。

And that is equal to, and my calculator is right here.

等於，我的計算器就在這裡。

So 45, take the sine, divide by 1.5, and take the arc sine of that.

所以 45，取正弦值，除以 1.5，再取其弧形正弦值。

And we have 28.1 degrees, so 28.1 degrees for theta sub 2.

我們有 28.1 度，所以 28.1 度為 Theta sub 2。

All right.

好吧

Now, if theta sub 2 is, and I'll write over here, is 28.1 degrees, how big is theta sub 3?

如果θ sub 2 是 28.1 度，那麼θ sub 3 有多大？

Hmm.

嗯

Well, again, we use the same principle we did over here.

好吧，還是那句話，我們用的原則和這裡一樣。

Notice that this line right here is parallel to this line right there.

請注意，這裡的這條線與那邊的這條線是平行的。

And then we have the line coming across as the beam of light, which means that theta sub 3 here is the same as this angle right there.

然後我們有一條線作為光束穿過，這意味著這裡的θ sub 3 與這裡的角度相同。

So this angle and this angle are equal to each other.

所以這個角和這個角是相等的。

Hmm.

嗯

So what is this angle equal to?

那麼這個角度等於多少呢？

Well, let's see here.

讓我們來看看。

We know that this angle from there to there is 90 degrees.

我們知道，從那裡到那裡的這個角度是 90 度。

So this is a 90 degree angle.

是以，這是一個 90 度角。

We also know that this here must be a 45 degree angle.

我們還知道，這裡必須是一個 45 度角。

Because these are opposite angles to each other.

因為它們互為對角。

So this angle is 45 degrees, that means this angle is 45 degrees.

所以這個角度是 45 度，也就是說這個角度是 45 度。

And that means that the sum of this angle plus theta sub 2 is also 45 degrees, which means if we take down the total, 45 degrees, and subtract from that theta sub 2, we get this angle right there.

這意味著這個角度加上θ sub 2 的總和也是 45 度，也就是說，如果我們把 45 度的總和減去θ sub 2，就得到了這個角度。

And since these are alternate interior angles, that means theta sub 3 must also be 45 degrees minus theta sub 2.

由於這些是交替內角，這意味著 theta sub 3 也必須是 45 度減去 theta sub 2。

So theta sub 3 is equal to 45 degrees minus theta sub 2.

是以，θ sub 3 等於 45 度減去θ sub 2。

Alright, and since theta sub 2 is 28.1 degrees, that means theta sub 3 is equal to 45 degrees minus 28.1 degrees.

好吧，既然 Theta sub 2 是 28.1 度，那就意味著 Theta sub 3 等於 45 度減去 28.1 度。

And so 45 minus 28.1 equals theta sub 3 is equal to, let's do that again, 45 minus 28.1 equals 16.9 degrees.

是以，45 減 28.1 等於 Theta sub 3，我們再來一遍，45 減 28.1 等於 16.9 度。

Alright, so now we have theta sub 3, which should allow us to find theta sub 4 again using Snell's Law.

好了，現在我們有了 theta sub 3，這樣就可以利用斯涅爾定律再次求出 theta sub 4。

So we have n1 sine of, oops, not n1, we're now going from 3 to 4, so let's call this n3.

所以我們有 n1 正弦，哎呀，不是 n1，我們現在是從 3 到 4，所以我們稱之為 n3。

So n3 sine of theta 3 is equal to n4 sine of theta sub 4.

是以，θ 3 的正弦值 n3 等於θ 子 4 的正弦值 n4。

We're looking for theta sub 4, so we can flip the equation around.

我們要找的是θ sub 4，是以可以將等式反過來。

Sine of n4 sine of theta sub 4 equals n3 sine of theta sub 3.

n4 正弦的正弦次 4 等於 n3 正弦的正弦次 3。

And so sine of theta sub 4 is equal to, when we divide both sides by n sub 4, we get n sub 3 divided by n sub 4 times the sine of theta sub 3.

是以，θ sub 4 的正弦等於，當我們用 n sub 4 除以兩邊時，我們得到 n sub 3 除以 n sub 4 乘以θ sub 3 的正弦。

And finally, we take the arc sine, and so we have theta sub 4 is equal to the arc sine or inverse sine of n3 over n4 times the sine of theta sub 3.

最後，我們取弧形正弦，是以，θ sub 4 等於 n3 乘以 n4 的弧形正弦或逆正弦乘以θ sub 3 的正弦。

And plug in the numbers, that's equal to the arc sine of n3, which is 1.5, n4, which is 1, times the sine of theta sub 3, which we said was 16.9 degrees.

輸入數字，等於 n3 的弧正弦值（1.5）、n4 的弧正弦值（1）乘以 theta sub 3 的正弦值（我們說過是 16.9 度）。

Alright, so we take the sine of that, we multiply it times 1.5, and we take the arc sine of that number, and we get 25.6 degrees.

好了，我們取其正弦，乘以 1.5，再取該數字的弧正弦，得到 25.6 度。

So theta sub 4 equals 25.6 degrees, and that's our answer.

是以，θ sub 4 等於 25.6 度，這就是我們的答案。

Alright, so you can see that the most difficult part of this problem is trying to find all the angles.

好了，你可以看到，這個問題最難的部分是試圖找到所有的角度。

And of course, it's getting very busy in here, but let's quickly recap how we did that.

當然，這裡現在非常繁忙，但讓我們快速回顧一下我們是如何做到這一點的。

We're given the angle of 45 degrees here, so we assume that's also a 45 degree angle over here.

這裡的角度是 45 度，所以我們假設這裡的角度也是 45 度。

This is a 90 degree angle.

這是一個 90 度角。

We draw the ray, the ray was coming in horizontally, so parallel to the bottom.

我們畫出射線，射線是水準射入的，所以與底部平行。

And so, if we look at this side, and we look at this line and this line, we can call these alternate interior angles, so they must be equal to each other.

是以，如果我們看這條邊上的這條直線和這條直線，我們可以稱它們為交替內角，所以它們一定相等。

That means that this angle must also be 45 degrees, because this is a 90 degree angle.

這意味著這個角度也必須是 45 度，因為這是一個 90 度的角度。

So we determine theta sub 1.

是以，我們確定了 Theta 子 1。

Then we calculated theta sub 2, which is the refracted angle by using Snell's Law.

然後，我們利用斯涅爾定律計算出折射角 theta sub 2。

Then we have to find out what theta sub 3 was equal to.

那麼我們必須找出 Theta sub 3 等於多少。

And to do that, we realize that this here is a 90 degree angle.

要做到這一點，我們要知道這裡是一個 90 度角。

This was a 45 degree angle, so we know that this whole thing here was a 45 degree angle, which means that this angle here is 45 degrees minus theta sub 2.

這是一個 45 度的角，所以我們知道這裡的整體是一個 45 度的角，這意味著這裡的角是 45 度減 θ 子 2。

And then if we draw this line here and this line there, we see that theta sub 3 and this angle are alternate interior angles, so they must be equal to each other right here.

然後，如果我們在這裡畫這條直線，在那裡畫這條直線，我們就會發現θ sub 3 和這個角是交替的內角，所以它們一定在這裡相等。

And since we knew what theta sub 2 was, we subtract from 45 degrees to find theta sub 3.

既然我們知道了 Theta sub 2 的值，那麼從 45 度減去 Theta sub 3 就可以了。

And then we use that in our equation right here to find theta sub 4.

然後我們將其應用到方程中，求出 theta sub 4。

And that's how you do that problem.

這就是解決這個問題的方法。