# Can voltage be measured manually?

I am a novice in the subject of electricity so please bear with me if my questions seem naive. What I basically wanted to know is that what is voltage in its essence and whether it can be measured manually (given a simple example, of course) rather than with a voltmeter. Like so far in my study of electricity, I think of voltage as something that is dictated by the electric field and columbs force, whose value depends on the shortest distance (since electric force is conservative) between points A and B (across which we are measuring the voltage). Please correct me if I am wrong! Like, lets say there is a bunch of positive charge at one side, and bunch of negative charge at other size. There is, of course, electric fields radiating to and from these charges and there is also force of attraction between these charges. So with these situation, I think that is particualr value of voltage is something that is specifically dependent on distance between these charges and the force you have to exert against the columbs force that already exists? So with this in mind, lets say that we have a simple example of couple of electrons and positive charges at a particular distance (whether it is in static electricity or dynamic electricity in a ciruit), can we calculate what the voltage may be between them by hand (by mathematical calculation, not physically) instead of using a voltmeter? I know this is long. But i feel that my understanding is either faulty or incomplete, so I needed second opinions. Thank you.

• If you know Ohm's law and the power formulas, you can calculate anything you want, but it still takes a meter. – BillDOe Jan 17 '17 at 20:43

Yes, with some amount of practice, one can tell 120V from 220-240V by using just hand's fingers. However, I would not recommend this method to anyone.

• I meant by mathematical calculation by "hand" or "manually" not literally. – TLo Jan 17 '17 at 20:38

Yes, it happens to be the case that if you knew exactly where all of the charges were in a system, and how strong those charges are, then you can calculate the voltage on an arbitrary point.

To do this, you simply divide the charge by how far away it is, and add all these numbers up, keeping in mind that you should cancel out charges of opposite signs.

If you are not measuring the charge by how strong its force is (for example as Gaussian units do), you will have to use a conversion factor to get the units of electric potential that are more natural for you (for example the SI unit of volts), however, this is just a fixed number that once you work it out once must be true for every other situation that you come into contact with.

For example, in the cgs system, where we measure charge in "statcoulombs" also called "esu", the force between two charged particles (in dynes, $\text{1 dyne} = 10^{-5}\text{ N}$) is equal to the charges of the two particles (in esu) divided by the number of centimeters between them. If I am sitting near two charges, one of which is $\text{2 esu}$ located $\text{5 cm}$ away, the other is $\text{-3 esu}$ and is located $\text{3 cm}$ away, the electric potential is $\frac25 - \frac33 = -0.6\text{ dyne}\cdot\text{cm}/\text{esu}$ (equivalently, you could say $-0.6 \text{ esu}/\text{cm}$ or call these "statvolts", $-0.6\text{ statV}$.)

This means that if I brought in a test charge of $\text{1 esu}$ from infinity, where the electric potential is 0, I would get $\text{0.6 dyne}\cdot\text{cm}$ of energy. You can look up that in terms of charge, $\text{1 esu}$ is about $3.33564\times10^{-10}\text{ C}$ in SI units and $\text{1 erg}=1\text{ dyne}\cdot\text{cm} = 10^{-7}\text{ J}$ exactly, so the conversion factor turns out to be that one "statvolt" is almost exactly $\text{300 V}$, so this is actually $\text{-180 V}$ in the more-familiar SI units. Note that only potential energy differences really exist, and therefore only voltage differences exist, and this voltage is being implicitly compared with the $\text 0V$ off at an infinite distance away from these charges!

Therefore, we know some example configurations where this calculation doesn't work, usually involving infinite lines or planes of charge with some charge density: you divide the charge into little charges spread out, and you work out that the voltage is always something like "infinity." This is because this way of working out the voltage above assumes that we're taking the voltage to be 0 at infinity, but these configurations require giving or taking infinite amounts of energy to move a particle out to infinity! In such cases we find it much more helpful to have the voltage be infinite out at infinity, and finite for all nonzero finite distances from the infinite wire/sheet. So we have to be a bit more crafty; we have to say "I know that voltage is not absolutely defined but only defined up to a constant, I am going to choose some place in real space to have 0 voltage, and then if I am at a distance $r$ from a charge $q$ that is a distance $L$ from that place, we will say that its actual electric potential upon me is $\frac qr - \frac qL,$ so that no matter what, if I am at this special place my voltage is 0." Then we can work out a voltage function for those cases as well, and it won't be infinite everywhere.

Yes, you can calculate the voltage by hand. Voltage is something that measures the pressure between two points, where the pressure force is the electric force. It's just like air pressure, in that voltage pressure can cause things to move. But if you don't complete the circuit, just like when you don't let air out of a balloon, the voltage just stays and no current flows.

But enough analogies. I don't know where you are in your education, but voltage comes from multiplying the average electric field by the distance between two points.

For example, I want to know the voltage between a point with an electric field magnitude of 10 N/C (newtons per coulomb), and another point with a field magnitude of 10 N/C as well. The points are 1mm away from each other. Before you can say what voltage is where, you choose reference ground, 0V. The first point, say, is 0V. Then we travel 1mm to the next point. The average electric field over this distance is obviously 10 N/C, and the distance is 1e-03 meters (1x10^-3 = 0.001 = 1mm). So you get the voltage between those points: 0.01V.

Now, that way of thinking will not get you far in practice. Be very careful about when this idea does and does not apply.

In reality, voltage is continuous, and has direction, so instead of multiplying averages, we take a vector integral of the electric field vector. It's the same thing, but we use a dot product instead of a scalar product, and it gives you an equation usually. Vector integrals like this are simple if you know vectors and calculus beforehand, but if you aren't familiar with those topics, you'll have a lot of ground to cover and it wouldn't be worth getting too far into, without properly educating yourself on the mathematics of both beforehand.

For more on the proper definition of voltage, look to the Wikipedia page on it. The integral definition comes up fairly quickly.

Before I go, I'd like to clear up one more thing. The electric field exists at all points in the entire universe - it is a concept, not a physical entity. (At least, not classically, and the new quantum electrodynamics might beg to differ, but that comes later. ) When we talk about the electric field of this and that, we mean their contribution to the electric field. But the electric field itself exists at all points, because it is a sum-total thing for all points.

As well, the electric field is not separate from the electric force. The electric field is the electric force. The E field is one way of representing a force in space, at a distance from whatever is causing it. For example, the electric field at some point will show what direction and how strong the force an electron would feel at that point. When people talk about the electric field moving through space, they really mean the electric force is moving through space. The field just changes as that force moves through it.

Ask yourself, then: What is the electric force?

And for that matter: What is the magnetic force?

The electric force exists wherever a charged object feels a force. So, in electrostatics we have charge A and charge B. Coulomb's law is the only way to determine the electric force, and from that electric force we can say that the electric field around the charges has changed. Once charges move, though, something new happens.

The magnetic force is the force that a charged particle feels, when it moves. Forget permanent magnets, we're talking pure magnetic force. Since all motion is relative, and charged bodies most fundamentally are particles like electrons, we say that the magnetic force is the result of charged things moving past each other. Which, you might notice, is not too different from the electric force, which is the force between charged particles if they were stationary.

Eventually, you see, thanks to topics like relativity, that the electric force and the magnetic force are the same phenomenon, with different points of view. You also find that an electric force causes two charged particles to move towards or away from each other, and then that movement causes a magnetic force, and every time they change position the electric force between them changes, and so on in a chain reaction. This is called near-field electromagnetic forces. See technologies like NFC (near-field communication). Eventually, in a totally amazing move, the electromagnetic forces can unbound themselves from their source, travelling not through molecules and charges, but rather moving as a force of its own. That's electromagnetic radiation, and it exists in the far-field.

These are the major points that, for whatever reason, nobody talks about in electromagnetics. From now on, if you see the phrase "electric field", conceptually think "electric force." Know that it is not the same as the electric force mathematically, because a field and a force are not the same. But they are so closely linked, that conceptually they are exactly the same. Even if the electric field is 0 at most places, it exists at all points.

Good luck with E&M! Sam Gallagher

• While it might make sense to use the word 'pressure' in the context of a flowing fluid analogy, it is a mistake to use it other than in that context. Voltage (electric potential) is neither force not pressure. It is energy per unit charge as measured against an agreed reference. – dmckee Jan 18 '17 at 3:22
• More specifically, voltage measures change in potential energy per unit charge. And in fact, one synonym for voltage is electric pressure, and not against reason. The reason air flows out of a compressed tank is the same reason current flows in a circuit. The failures of the analogy are not lost on me, but pedagogically it is a much more sound method of describing voltage than using maths, which leave much to be desired. Frankly, I assume this knowledge is used in tandem with typical high school E&M education. – Sam Gallagher Jan 18 '17 at 14:18