Some things that physicists study are harder to measure than others. If you’re looking at the acceleration of a free falling object, it’s pretty easy. Just take a ball and drop it—measure the height and time and you get some nice data. You can even measure the gravitational field without too much difficulty.

But what about electric fields? How do you measure those? It’s not as simple, but there is something you can do. The idea is to use a sheet of electrically conducting paper to sort of map out the field. It mostly works, but there are some problems. So let’s go over this whole thing.

What Is An Electric Field?

To understand what an electric field is, it helps to start with the analogous idea of a gravitational field. Imagine you’re holding a ball with a certain mass (**m**). There is a downward gravitational force (**F**) on this ball, which equals its mass times a constant, **g**, where **g** = 9.8 newtons per kilogram.

If the ball has a mass of 1 kilogram, the gravitational force would be 9.8 newtons. If you replaced that with a mass of 10 kilograms, the gravitational force would be 98 newtons. What stays the same for these two masses? The constant **g**—we call this the gravitational field. It’s essentially the *force per unit mass* near the surface of Earth:

Now let’s do something similar with some electric charges. An electric charge could be a particle, like a proton or an electron, or it could be an object with nonzero net charge—like a sock when it comes out of the dryer. Although the electric interaction between two charges is usually much stronger than the gravitational force, you normally don’t notice it, because there are a similar number of positive and negative charges to mostly cancel the interaction.

Then what about those socks in the dryer? Yes, two socks can be charged up and interact with each other, but socks are too big to be useful here. What if you could get a single *point-like* charge by itself? A point charge is so small that the size doesn’t even matter. An electron is a perfect example. Suppose it has a charge of 1 coulomb (which is huge, but don’t worry about that now). If this charge experienced an electric force of 1 newton, then we could say it is in the region of an electric field. The value of this electric field can be calculated as:

In this expression, the vector **E** is the electric field, **F** is the electric force, and **q** is the electric charge. Just like the gravitational field is the force per unit mass, the electric field is the *force per unit charge*. So, the electric field gives us a way to describe the “region of electrical influence” around a charge or charge distribution. Trust me, this stuff is useful!

Electric Potential

Next, let’s consider the change in electric potential. The best way to think about potential energy is to return to our analogy with gravity. If you take that 1 kg ball and lift it up 1 meter, it takes work (9.8 joules worth to be exact). This increases the gravitational potential energy of the ball.

The same thing can be done with an electric field and an electric force. Instead of going into a full derivation (which involves the work-energy principle), I’m just going to define it. If you have a constant electric field, then the change in electric potential would be:

In this expression, **ΔV** is the change in the electric potential, **E** is the magnitude of the electric field, and **s** is the distance over which you move. This is actually the change in electric potential energy per unit charge, but we often are lazy and just call it “potential” or “voltage,” since it’s measured in units of volts. Whatever you call it, just don’t forget that you are always dealing with a *change* in potential.

One final thing. If you know how the electric potential is changing in space, you can find the electric field with the following relationship:

This says that the *x* component of the electric field (i.e., in the *x* direction) depends on the way the electric potential changes with location. If you moved in the *y* direction instead, you would be able to figure out the *y* component of the electric field. Remember that the electric field is still a vector. One value doesn’t give the full story.

But here’s the thing. You can’t really measure the electric field directly—but you *can* measure the electric potential difference. This is what a voltmeter does. So, let’s do an experiment and make some measurements. In doing this, I think you can get a better understanding of the relationship between electric potential difference and the electric field.

A Classic Experiment

Here’s the setup. It uses this electrically conducting paper with some lines drawn in conducting ink (the two silver lines you see in the picture) to represent electrically charged plates. (They are just lines because it’s two-dimensional paper—I wanted the 3D paper, but it was on back order).

The device on the left with the red cables is a power supply. It’s essentially like a variable battery—in this case, it’s set to 6 volts. I have the positive output going to the conducting plate on the right of the paper and the negative to the one on the left. These two plates have an electric potential difference of 6 volts.

The device on the right with the black cables is the voltmeter (technically, it’s a multimeter, since it measures more than just electric potential). Notice that you need *two* wires to measure a change in electric potential. I can put one of these wires on the negative conducting plate and the other can move to different locations on the paper. That way I can measure the electric potential at those points with respect to the negative plate.

(Oh, just a quick comment. When you calculate the theoretical electric potential, it’s common to do so with respect to a point at infinity. However, my cable isn’t quite that long, so I just used the negative plate as my reference point.)

OK, let’s get some data! Since the conducting paper has marks every centimeter, I can just move the positive part of the voltmeter to different spots and record the voltage values and locations using (*x*, *y*) coordinates. From that, I can make the following contour map:

That’s actually pretty awesome. But what does it mean? These contours are similar to the lines on a topographical map. For the topo map, each line consists of a bunch of points that are at the same altitude (above some fixed point—maybe sea level). Similarly, the lines on the electric potential plot are made of a bunch of points at the same electric potential (with respect to the negative plate). We call these equipotential lines.

In this potential plot, the two conducting plates are at the top and bottom of the graph. You can see that moving parallel to these conducting plates is mostly moving along an equipotential line. Looking at these lines can give you an idea about the electric field in this region. Perhaps the best way to understand it is to think of it like a topographical map. That would make this a hill with the top line at 6 “meters” (instead of volts) and the bottom line around 0 meters. Since the contour lines are fairly evenly spaced, it’s mostly a straight downhill. In terms of electric field, this would make a constant electric field pointing “downhill.”

But how about a numerical value for the electric field between these conducting plates? If I just go straight down the middle from one plate to the other, I can get electric potential values for different y values. Here’s what that looks like:

Remember the relationship between the electric field and the potential. The electric field is the negative of the change in potential divided by the change in position. If you plot potential vs. position, this is the same as the slope. Notice that the plot above is a linear function. This means the slope, and thus the electric field, is constant. From the slope, I get a constant electric field of 0.713 volts per cm (0.00713 V/m). Oh, 1 V/m is the same as a newton per coulomb. Both are equivalent units for the electric field.

But wait! The electric field is related to the electric force, and that means it should be a vector. The value calculated above is from the slope, so it’s just a scalar value. Well, there’s an easy fix for that. Since I plotted the potential with respect to the *y* position, this gives me the *y* component of the electric field. To find the *x* component, I’d also need to plot electric potential in that direction.

But in this case, the potential really doesn’t change much in the *x* direction. This means the x component of the electric field would be zero V/m. Honestly, that’s the nice thing about these parallel conducting plates—they make a constant electric field in one direction.

Why Do We Need the Paper?

So, that’s a quick introduction to electric fields and electric potential difference. Now for an answer to an important question that you didn’t ask:

Suppose I take a 9-volt battery and use some wires to connect the terminals to two parallel strips of aluminum foil separated by a distance of 10 cm—without any paper. Could I repeat this experiment to calculate the electric field between these plates?

The answer is no. I mean, it *should* work. The theory is that you have a change in potential across the two pieces of aluminum and there is a change in distance. Since you have two parallel plates, the electric field should be fairly constant. But it won’t work. If you take your voltmeter and connect one probe to the negative strip and put the other one right in the middle, it should read 4.5 volts. Instead it will read zero volts.

The problem is the voltmeter! It only *sort of* measures the change in electric potential. What it’s really doing is measuring an electric current. If you know the current going through a known resistor (inside the voltmeter), you can calculate the change in electric potential. You don’t need a very large current, but you need a current. For the two sheets of aluminum foil, there’s just not enough electric current going through the air to let the voltmeter get a reading.

This is why we use electrically conducting paper. When a potential difference is applied to the two parallel conducting plates, there is an electrical current going through the paper. When you put the voltage probe on the paper, it’s measuring a current to calculate the voltage. That’s why it works.

Of course, it’s not a perfect setup. We want to explore the electric field between two parallel plates, and we don’t have plates, we just have lines representing plates. That’s because the paper is two-dimensional, not 3D like real life. For a situation like this, it’s not a big deal, because you still get constant electric fields. But it’s not going to be the same as a real 3D point charge. And there’s another problem—edge effects. If you get closer to the edge of the paper, the edge will change the direction and magnitude of the current. It adds an interesting boundary condition, but it also makes it not like real 3D electric charges.

Still, it’s a classic experiment that lets you actually calculate an electric field—I kind of like it.

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