# How Do Object Distance and Focal Length Affect Depth of Field?

Note: this is Part 2 of a series on depth of field. Click here for part 1, which explains the relationship between aperture size and depth of field.)

Besides using aperture size, photographers can also control depth of field by changing the camera-to-object distance or the lens focal length. Specifically:

• For a fixed focal length and aperture size, increasing the object distance will increase the depth of field.
• For a fixed object distance and aperture size, increasing the focal length will decrease the depth of field.

These two effects are closely related, and the quest to understand them offers a great example of how we can use equations to augment our intuition while solving relatively complicated problems. Read on if you want to learn more!

# The Secret Lives of Electrons

It’s not much of an exaggeration to say that electrons form the backbone of the modern world. They are our workhorses—they bring energy from power plants to our homes and factories. They are our couriers—they carry information through circuits in our computers. Some day, they will be our providers—as the central elements in photovoltaic solar cells, they will capture the sun’s energy for our use. These current and future electronic devices rely on very precise answers to the question What do electrons do inside materials? Out of technological need and intellectual curiosity, condensed matter physicists have spent over a century discovering better and better answers—and they continue to do so today. In this post, I will try to give you a little glimpse into the form, beauty, and utility of such an answer.

# Non-visual Vistas, Non-spatial Travels

As some of you know, I am currently a graduate student in an experimental condensed matter physics research group. I’m sure that physics graduate school is quite mysterious to people who aren’t in it, and one of my goals for this blog is to explain what I do there and why I choose to do it. I hope my thoughts can be useful to both prospective physics students and to my non-physicist friends and family.

This certainly is a big task, and there’s no chance of me completing it in a single post. Instead, I’d like to start by telling you about the “a-ha moment” when I suddenly realized that condensed matter physics was interesting. It happened on Friday, March 6, 2009, sometime between 3:20 PM and 4:40 PM. It was the spring semester of my sophomore year of college; I was sitting in my thermal physics class, and we were learning about phase diagrams. In particular, my professor showed us the phase diagram for H2O, which I will now show you:

There is a clear scientific meaning to this picture. Each point on the diagram represents a pressure value and a temperature value; the diagram itself tells us whether H2O will be a solid (ice), liquid (water), or gas (steam) at that particular combination of temperature and pressure. The thick black lines show the locations of phase transitions where H2O changes from one phase to another.

Until this point in my life, I had always felt a sort of impersonal interest in physics. But something suddenly “clicked” when I saw this phase diagram. I had a big realization—that scientific facts really only come alive when interpreted by a human and related back to human experience. I don’t know why this happened. It may have been that I had finally matured enough to realize this, or I may have just needed something special to come along and shake me out of my “scientific stupor.” At any rate, I had an epiphany, which I would now like to share with you.

The first thing to notice is that the phase diagram contains all of our common experiences with H2O. Atmospheric pressure at the surface of the Earth is around 1 bar, which you can find as a horizontal line on the graph. As we travel along this line from low temperature to high temperature, we see that H2O changes from ice to water at 273 K (equivalent to 0°C or 32°F) and from water to steam at 373 K (equivalent to 100°C or 212°F).

The second thing to notice is that the phase diagram contains so much more than all of our common experiences with H2O! People live at pressures near 1 bar and only deal with temperatures between 0-100°C, but H2O exists in a much wider range, and it does many other things besides melt and boil. Let’s discuss three of them:

1. At low pressures—say, 0.001 bar—water does not exist at any temperature. H2O goes straight from ice to water vapor in a process called sublimation.
2. There is a single point on the diagram where ice, water, and steam can all coexist. (I marked it with a red dot.) It is located at a pressure of 0.006 bar and a temperature of 0.01°C. Imagine visiting a planet whose atmospheric pressure was 0.006 bar. Water would only exist if the temperature was exactly 0.01°C. If you tried to melt ice, you would find that it simultaneously turned into both water and steam! (The picture below shows solid argon doing this at atmospheric pressure.)
3. There is another special point where the dividing line between liquid and gas simply disappears. (I marked it with a blue dot.) It is located at a pressure of 221 bar and a temperature of 374°C. At greater temperatures and pressures, there just isn’t a difference between liquid and gas. An interesting consequence of this is that you could turn water into steam without boiling it if you could first raise its pressure to 230 bar, then increase its temperature to 380°C, and finally decrease its pressure back down to 1 bar.

In one sense, H2O’s phase diagram vividly demonstrates how narrow the human experience is. (Well, it demonstrates one aspect of a much larger narrowness.) We think that ice melts into water, but sometimes it sublimates into steam, and sometimes it does both. We think of water and steam as totally different, but under some circumstances, they aren’t. Living at a single pressure is like always looking in the same direction. Learning about this phase diagram is like suddenly turning your head to the left and right. A whole new vista becomes visible.

In another sense, the diagram shows that even boring everyday substances can become interesting when they’re subjected to unusual conditions. People like to travel because new and different surroundings are fun and exciting; physics teaches us that changing spatial coordinates is not the only way to achieve this effect. Moving to exotic temperatures and pressures is a whole new way to travel.

We can ask a lot of questions about phase diagrams. For example:

• Do all substances have similar phase diagrams?
• If not, what changes between substances? Are there certain groups that have similar phase diagrams?
• Why are different phases stable in different regions?
• Can we predict the phase diagram for a particular substance?
• Are there other kinds of phases besides solid, liquid, and gas?
• Are there other parameters besides temperature and pressure that can change a substance’s phase?
• Are there different kinds of phase transitions?

Answers to these questions can satisfy out natural curiosity about the physical world and lead to new technological applications. One goal of condensed matter physics is to answer them as fully as possible.

Phase diagrams didn’t point me unambiguously towards condensed matter physics, but they did open my mind to it. They taught me a new way of thinking, and they opened my internal eyes to viewless vistas. Ever since that day in March 2009, I have had a soft spot for them in my “physics heart,” and I hope you can at least partially appreciate why.

## References

• The data for the melting and sublimation lines of water are from “International Equations for the Pressure along the Melting and along the Sublimation Curve for Ordinary Water Substance”, J. Phys. Chem. Ref. Data, Vol 23, No 3, 1994.
• The data for the boiling line of water are from IAPWS formulation for Industrial Use, 1997.
• The picture of argon ice is from http://en.wikipedia.org/wiki/File:Argon_ice_1.jpg.

# Why Does a Small Aperture Increase Depth of Field?

Photographers know that decreasing the aperture size on their camera will produce an image with a larger depth of field. Although it’s possible to take a great picture without understanding why this is true, it doesn’t hurt to know! All we need to figure it out is a little bit of geometry and physics, plus a little knowledge about how a camera works. This is a really fascinating topic for me; it shows how a single topic can have many different levels of understanding.

# Of Temples and Table Salt

Have you ever seen a diagram of a substance’s atomic structure (like the one below, for table salt) and wondered where it came from? If you haven’t, I’ll try to quickly explain why the existence of such a picture might be a mystery:

• Typical atomic sizes are between 30 and 300 picometers. A picometer is one thousandth of a nanometer, which is one millionth of a millimeter. Atoms are really small!
• Typical interatomic spacings in solids are bigger, but not by much; they are usually between 0.1 and 1 nanometers. So at the very least, we need to be able to detect things that are about 1 nanometer in size to determine the atomic structure of a molecule or material.
• The physical size of the average pupil prevents human eyes from seeing anything much smaller than a hair, which is about 0.1 millimeters wide– far larger than a nanometer! (Go here for an explanation.)
• In fact, the large wavelength of visible light prevents any standard optical device (including eyes and microscopes) from seeing anything smaller than a few hundred nanometers.
• X-rays have wavelengths comparable to interatomic spacings, but it is nearly impossible to build lenses for x-rays. Thus, x-ray microscopes don’t really exist.

So how do we know what anything looks like on such a small scale? Well, people have actually invented a whole bunch of clever methods for seeing very tiny things. I would love to talk about all of them, but for now I am just going to focus on one of the oldest and most widely-used techniques: x-ray diffraction. X-ray diffraction is a tool for determining the structure of a crystal, which is a solid material that has a repeating atomic structure. It’s true that not every material has a repeating structure, but many do. What’s even better is that many molecules can be tricked into growing in a crystalline form; for example, proteins can be stacked into periodic arrays. This allows x-ray diffraction to determine their molecular structures. This information is invaluable to physicists, chemists, biologists, doctors, and pharmaceutical companies. (Here is a big ol’ pile of protein structures.)

# How To Solve Doppler Effect Problems

If you’re having difficulty solving classical Doppler shift problems on your physics homework, you’re not alone! Many textbooks portray the Doppler shift formula in confusing ways that obscure real understanding. It is often stated as either several different formulas (one formula for a moving source and stationary observer, one formula for a stationary observer and moving source, etc) or one formula with a cruel and impossible-to-remember sign convention (for example, “source velocity is positive when the source moves in the direction of the emitted wave”).

I prefer to use what one of my past professors used to call the “Method of Thinking” to solve Doppler shift problems. (This is just one use of a more general Method of Thinking, in which you simply think about a problem to find the answer.) The basic ingredients are one easy-to-remember “incomplete formula,” two simple rules, and an instruction to think:

1. $\displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave} \blacksquare |v|_{observer}}{|v|_{wave} \blacksquare |v|_{source}}$, where the velocities are relative to the medium and $\blacksquare$ stands for either plus or minus. The following two rules determine which sign to choose. (The formula is written in a general way to emphasize that the Doppler shift formula applies to many kinds of waves, but it is good to think of sound as a representative example. In that case, the wave speed is the speed of sound and the medium is air.)
2. If the source and receiver move towards each other (and one is stationary), the observed frequency increases.
3. If the source and receiver move away from each other (and one is stationary), the observed frequency decreases.
4. Think!

# Examples

1. Suppose the observer is stationary with respect to the medium and the source moves towards it. Then $v_{observer} = 0$ and Rule 2 tells us that the frequency must increase. So the correct formula to use is $\displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave}}{|v|_{wave} - |v|_{source}}$. We choose the minus sign because shrinking the denominator of a fraction makes it grow.
2. Suppose the source is stationary with respect to the medium and the source moves away from it. Then $v_{source} = 0$ and Rule 3 tells us that the frequency must decrease. So the correct formula to use is $\displaystyle f_{observed} = f_{emitted} \frac{|v|_{wave} - |v|_{observer}}{|v|_{wave}}$. We choose the minus sign because shrinking the numerator of a fraction makes it decrease.
3. Suppose both the source and observer are moving with respect to the air. This situation involves a little bit more Rule 4 than the first two examples. The trick is to imagine a third entity (call it “The Phantom”) at rest with respect to the air. The Phantom listens to the sound emitted by the source (call this “Stage 1”) and emits an exact copy to the observer (call this “Stage 2”). Thus, The Phantom is both an observer and a receiver. In Stage 1, The Phantom hears a frequency given by $\displaystyle f_{phantom} = f_{emitted} \frac{|v|_{wave}}{|v|_{wave} \blacksquare |v|_{source}}$, where the sign is determined as in the previous examples. In Stage 2, the observer hears a frequency given by $\displaystyle f_{observed} = f_{phantom} \frac{|v|_{wave} \blacksquare |v|_{observer}}{|v|_{wave}}$, where the sign is determined as before. Inserting the equation for the Phantom frequency into the equation for the observed frequency gives the correct formula. Note that the top of the Phantom frequency fraction will cancel with the bottom of the observed frequency fraction.
4. If the source and the observer are at rest relative to the ground, there can still be a Doppler shift if the medium is moving. This is equivalent to the previous case. (This means that on a windy day, your voice could appear to be lower or higher.)

It is not useful to remember the specific equations; instead, try to learn the process that leads to them. This allows you generate formulas as you need them from a few basic concepts, instead of cluttering your head with multiple and/or confusing equations. Note: this is a good example of the power of the physicist’s mode of thinking.

Continue forward to learn where Rules 1, 2, and 3 come from!