Geometric Optics
AP Physics 2 Topic 13 9:32 English narration · English + 中文 subtitles burned in
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Put a straight pencil into a glass of water, and it looks broken.
把一支笔直的铅笔放进水杯里，它看上去像断了。
Look into the bowl of a shiny spoon, and your face is upside down.
看看亮亮的勺子内侧，你的脸就是倒着的。
Nothing bent, and nothing moved.
其实什么都没有弯，也没有东西移动。
Light simply changed direction — once at the water, once at the metal.
只是光改变了方向——一次在水面，一次在金属面。
Learn the two rules behind that, and you can predict every mirror, every lens, and every camera.
学会这背后的两条规则，你就能预测每一面镜子、每一个透镜、每一台相机。
This is Unit Thirteen: geometric optics.
这是第十三单元：几何光学。
Here we forget that light is a wave, and draw it as straight lines called rays.
在这里我们先不管光是一种波， 而是把它画成一条条直线，叫做光线。
Two rules — reflection and refraction — are enough to build every mirror, every lens, and every image they make.
只用两条规则——反射和折射—— 就足以解释每一面镜子、每一个透镜，以及它们成的每一个像。
First, the model.
先说模型。
A light ray is a straight line that points the way the light travels, at right angles to the wavefront.
光线是一条直线，指向光传播的方向，并且与波前垂直。
A ray diagram shows the path before and after the light meets matter — one picture, most of the marks.
光路图画出光遇到物质之前和之后的路径——一幅图，就是大部分的分数。
But rays have limits: they cannot explain interference, or how light spreads through a gap. That needs the wave, in Unit Fourteen.
但光线模型有局限：它无法解释干涉，也无法解释光通过窄缝时的扩散， 那需要波动模型，在第十四单元。
The closest thing to one ray is a laser: one colour, one direction, one narrow beam.
现实中最接近单条光线的，是激光： 一种颜色、一个方向、一道细束。
Rule one: reflection.
规则一：反射。
Draw the normal first — the line at right angles to the surface, where the light lands.
先画出法线——在光落下的那一点，与表面垂直的那条线。
Every angle in this unit is measured from that normal, never from the surface.
本单元中所有角度都从法线量起，绝不从表面量起。
The law of reflection says the angle of reflection equals the angle of incidence, on the other side of the normal.
反射定律说：反射角等于入射角，位于法线的另一侧。
On a smooth surface the normals all point the same way, so the rays leave together and the image is sharp: specular reflection.
在光滑表面上，各点的法线方向一致，光线整齐地一起离开，像很清晰，这叫镜面反射。
On a rough surface the normals tilt, so light scatters: diffuse reflection.
在粗糙表面上，法线逐点倾斜，于是光向各个方向散开，这叫漫反射。
Now bend the mirror.
现在把镜子弯曲起来。
A concave mirror curves inward: parallel rays come back to one point, the focal point, halfway between the mirror and its centre of curvature.
凹面镜向内弯：平行光线反射后回到同一点， 这一点就是焦点，位于镜面与曲率中心的正中间。
A convex mirror curves outward: the rays spread, and only appear to come from a focal point behind the glass.
凸面镜向外弯： 光线反射后散开，只是看上去像是从镜子后面的一个焦点发出的。
A plane mirror is the flat limit; your image sits as far behind the glass as you stand in front.
平面镜是平直的极限；你的像在镜后的距离，正好等于你站在镜前的距离。
Two words matter here.
这里有两个词很重要。
A real image forms where rays truly cross, so a screen can catch it.
实像形成在光线真正相交的地方，可以用屏幕接住。
A virtual image forms where they only appear to cross.
虚像形成在光线只是看起来相交的地方。
One equation locates the image: one over the image distance, plus one over the object distance, equals one over the focal length.
有一个公式可以定出像的位置：像距的倒数，加上物距的倒数，等于焦距的倒数。
The magnification is the image height divided by the object height, and also minus the image distance divided by the object distance.
放大率等于像高除以物高，也等于负的像距除以物距。
Signs carry the physics.
符号承载着物理意义。
A positive image distance puts the image in front of the mirror, so it is real; a negative one puts it behind, so it is virtual.
像距为正，像在镜前，也就是光真正到达的一侧，所以是实像； 像距为负，像在镜后，所以是虚像。
A concave mirror has a positive focal length, a convex mirror a negative one.
凹面镜的焦距为正，凸面镜的焦距为负。
Let's use it.
我们来用一用。
A concave make-up mirror has a focal length of twelve centimetres, and your face is eight centimetres in front of it.
一面凹面化妆镜的焦距是十二厘米，你的脸在镜前八厘米处。
Pause here and work it out.
先暂停，自己算一算。
Ready?
好了吗？
One over the image distance equals one over twelve, minus one over eight, which is minus one over twenty-four.
像距的倒数等于十二分之一减去八分之一， 也就是负的二十四分之一。
So the image distance is minus twenty-four centimetres: negative, so virtual, behind the mirror.
所以像距是负二十四厘米：负号表示虚像，在镜子后面。
The magnification is minus the image distance divided by the object distance, which is plus three: upright, and three times bigger.
放大率等于负的像距除以物距，等于正三：正立，而且放大三倍。
That is a shaving or make-up mirror used exactly this way.
剃须镜或化妆镜就是这样用的。
Rule two: refraction.
规则二：折射。
Light travels more slowly inside glass or water than in empty space.
光在玻璃或水中传播的速度，比在真空中慢。
So when a ray crosses a boundary at an angle, one edge of the beam slows before the other, and the whole ray turns.
所以当光线斜着穿过分界面时，光束的一边比另一边先慢下来，整条光线就转了方向。
Watch it bend toward the normal going in, and back away from the normal coming out.
看它进入时向法线偏折，离开时又向远离法线的方向偏回去。
Now put numbers on it.
现在给它加上数字。
The index of refraction of a material is the speed of light in a vacuum divided by its speed in that material.
材料的折射率，等于真空中的光速除以光在该材料中的速度。
Water is one point three three; glass is about one point five.
水是一点三三，玻璃约为一点五。
A bigger index means slower light, and stronger bending.
折射率越大，光越慢，偏折也越强。
Snell's law ties the two sides of the boundary together: the first index times the sine of the first angle equals the second index times the sine of the second angle.
斯涅尔定律把界面两侧联系起来：第一个折射率乘以第一个角的正弦， 等于第二个折射率乘以第二个角的正弦。
Three consequences.
有三条结论。
Going into the denser material, the ray bends toward the normal.
进入较密的介质时，光线向法线偏折。
Coming out into the faster one, it bends away.
出到较快的介质时，向远离法线偏折。
And a ray along the normal is not bent at all.
而沿着法线入射的光线，完全不偏折。
A worked example.
来看一道例题。
A ray in air hits the surface of water at forty degrees to the normal.
一条光线在空气中以与法线成四十度的角射到水面。
Air has an index of one point zero zero, water one point three three.
空气的折射率是一点零零，水是一点三三。
Snell's law gives the sine of the second angle as one point zero zero divided by one point three three, times the sine of forty degrees. That is zero point four eight three.
由斯涅尔定律，第二个角的正弦等于一点零零除以一点三三，再乘以四十度的正弦， 结果是零点四八三。
Take the inverse sine, and the refracted angle is twenty-nine degrees.
取反正弦，折射角是二十九度。
Check that it makes sense: twenty-nine is smaller than forty, so the ray bent toward the normal, just as it must going into a denser material.
检查一下是否合理：二十九小于四十，说明光线向法线偏折了， 进入较密的介质时正应如此。
Now send the light the other way — from inside the glass, out toward the air.
现在让光反过来走——从玻璃内部射向空气。
As the inside angle grows, the escaping ray bends further from the normal and gets dimmer.
随着内部角度变大， 射出去的光线离法线越来越远，也越来越暗。
At one special angle it lies flat along the surface.
在某一个特殊的角度，它会贴着表面走。
Push past that angle, and nothing escapes at all.
再超过这个角度，就完全没有光射出去。
That angle is called the critical angle.
这个角度叫做临界角。
There the refracted ray leaves at ninety degrees to the normal, flat along the boundary.
在临界角处，折射光线沿着与法线成九十度的方向射出， 也就是贴着界面走。
Put ninety degrees into Snell's law, and the sine of the critical angle becomes the second index divided by the first.
把九十度代入斯涅尔定律， 临界角的正弦就等于第二个折射率除以第一个折射率。
For glass out to air that is one point zero zero over one point five zero, so the critical angle is forty-two degrees.
对于玻璃射向空气，就是一点零零除以一点五零，所以临界角是四十二度。
Beyond it, no light is transmitted — and this can only ever happen going from a slower material into a faster one.
超过这个角度就没有光透射出去——而且这只会发生在从较慢的介质射向较快的介质时。
This is how an optical fibre carries a phone call across an ocean: the ray is too steep to escape, so it bounces for kilometres.
光纤就是这样把一通电话送过大洋的：光线太陡逃不出去，于是来回反射走上千米。
A lens is refraction used twice: once at the front surface, once at the back.
透镜就是把折射用了两次：一次在前表面，一次在后表面。
Here is the standard construction, and it takes only two rays.
下面是标准作图法，只需要两条光线。
From the top of the object, draw a ray parallel to the axis; after the lens it must pass through the focal point on the far side.
从物体顶端画一条平行于主轴的光线， 经过透镜后它必定通过另一侧的焦点。
Then draw a ray straight through the centre of the lens, which is not bent.
再画一条光线，直接穿过透镜中心，完全不偏折。
Where they cross is the top of the image.
这两条光线的交点，就是像的顶端。
Two shapes of lens behave in opposite ways.
透镜有两种形状，作用正好相反。
A converging lens is thicker in the middle: parallel rays are pulled together to a real focal point behind it.
凸透镜中间厚：平行光线被会聚到它后面的一个实焦点。
A diverging lens is thinner in the middle: parallel rays spread, and only appear to come from a focal point in front of it.
凹透镜中间薄：平行光线被发散开，看上去像是从它前面的一个焦点发出的。
The three principal rays are the ones worth drawing: parallel in and out through the focus, straight through the centre, and in through the near focus and out parallel.
值得画的是三条特殊光线：平行入射、出射后过焦点的那条；直接穿过透镜中心的那条； 以及经过近侧焦点、出射后平行于主轴的那条。
And remember one thing: a diverging lens always makes a virtual, upright, smaller image.
记住一点：凹透镜永远只成正立、缩小的虚像。
Watch what happens as the object moves.
看看物体移动时会发生什么。
Far away, the image is small, upside down, and close to the focal point.
物体很远时，像很小、倒立，而且靠近焦点。
Slide the object toward the lens, and the image runs away from the lens and grows.
把物体向透镜移近，像就会远离透镜并且变大。
That one movement is the whole behaviour of a camera lens and of a projector.
这一个动作，就概括了照相机镜头和投影仪的全部行为。
Now the numbers.
现在来算数字。
An object stands thirty centimetres from a converging lens of focal length ten centimetres.
一个物体位于焦距为十厘米的凸透镜前三十厘米处。
The thin lens equation has the same form as the mirror one.
薄透镜公式的形式与球面镜相同。
One over the image distance equals one over ten, minus one over thirty, which is two over thirty, so the image distance is fifteen centimetres.
像距的倒数等于十分之一减去三十分之一， 也就是三十分之二，所以像距是十五厘米。
It is positive, so the image is real, on the far side of the lens.
它是正的，说明是实像，在透镜的另一侧。
The magnification is minus fifteen divided by thirty, or minus zero point five: upside down, and half size — what a camera does to the world.
放大率等于负十五除以三十，即负零点五：倒立，大小是原来的一半—— 这正是照相机对世界所做的事。
Most marks lost in this unit are sign mistakes, so keep one small table in your head.
这个单元里丢掉的分数，大多来自符号错误，所以脑子里要有一张小表。
A positive focal length means converging: a concave mirror, or a convex lens; a negative one means diverging.
焦距为正表示会聚：凹面镜，或者凸透镜；焦距为负表示发散。
A positive image distance means a real image, on the side the light actually travels to; a negative one means a virtual image, on the same side as the object.
像距为正表示实像，位于光真正到达的那一侧；像距为负表示虚像，与物体在同一侧。
And one case is worth knowing by heart: put the object closer than the focal length of a converging lens, and the image comes out virtual, upright and enlarged.
有一种情况值得背下来：把物体放在凸透镜的焦距以内， 成的像就是虚像、正立、放大的。
You are holding a magnifying glass.
你手里拿的就是一只放大镜。
Three habits that save marks.
三个能保住分数的习惯。
First, every angle in this unit is measured from the normal, never from the surface, so draw the normal before anything else.
第一，本单元中所有角度都从法线量起，绝不从表面量起， 所以先画法线，再画别的。
Second, total internal reflection only happens going from a slower material into a faster one, above the critical angle.
第二，只有当光从较慢的介质射向较快的介质， 并且超过临界角时，才会发生全反射。
Third, write your sign convention down before you substitute, then read the sign of your answer as physics: a negative image distance is a virtual image, and a negative magnification an inverted one.
第三，代入数字之前先写下你的符号约定， 然后把答案的符号当作物理来读：像距为负就是虚像，放大率为负就是倒立的像。