Natural Constants: µ and ε (Vacuum Permeability and Permittivity)

Updated on April 24, 2020
Nikola Tesla basking in his high voltage laboratory.
Nikola Tesla basking in his high voltage laboratory. | Source

Part of an overview of the physics and mathematics pertaining to naturally occurring constants. In this chapter: µ0 and ε0.

Don't try this at home.
Don't try this at home. | Source

Have you had enough coffee today?

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Bar magnets showing lines of magnetic force with iron filings.
Bar magnets showing lines of magnetic force with iron filings. | Source

Vacuum Permeability

Often referred to as the constant of vacuum permeability, µ0 defines the value of magnetic permeability in a classic vacuum. To better understand magnetic permeability, consider pouring coffee into an large imaginary mug that has a vacuum within it. It is fair to assume that the coffee will fill the space contained by the mug (as it would with only air in the mug). If we instead filled that mug with paper towels something very different would happen. Depending on how densely we pack the paper towels into the mug, some or most of the coffee will be absorbed. If we additionally assume that we have infinite coffee (you’ve pulled multiple all-nighters studying magnetic permeability), then we can say that the paper towel was reasonably saturated with coffee based on how much was stuffed into the space; however, the constant flow of coffee is unaffected and will continue to flow through the paper towel. Lastly, we cover the top of the mug with a metal mesh strainer. When we pour coffee into the mug this time, it flows through the strainer (like it did through the paper towel), yet none of it is absorbed. Therefore, different materials “accept” or respond to the coffee differently.

Case in Point

Such is the case with magnetic permeability, which suggests that materials respond differently in the presence of a magnetic field: like absorbing coffee, materials may gain their own magnetic field. Magnetic permeability measures the change of the material, induced magnetic field relative to the source magnetic field. To continue our analogy, take the wad of soaking brown paper towel, and compare it to the coffee pot from which you are pouring. Though we have infinite coffee coming out of it, it does have a finite volume (in this imaginary situation). How much coffee can that sop-wad hold in relation to the coffee pot? I hazard to label this “coffee permeability.” Finally, vacuum permeability is the effect of the magnetic field in a vacuum. So all absolute material permeability may be compared to vacuum permeability to obtain a relative permeability.

Frog levitating due to saturation in ferro-liquid.
Frog levitating due to saturation in ferro-liquid. | Source


Less a derivation than an extended definition, µ0 was discovered as a proportionality constant in calculating current. Since currents were not yet defined when it was discovered, scientists did themselves a favor and gave it an exact value which was derived from a calculation of current and resulting magnetic force. Ampere’s force law, as we know it today, states the following:

Keep in mind, µ0 was only a constant of proportionality then. The new equation uses µ0/2π instead of 2kA. Determined to define both units of current and the constant of proportionality, scientists held two wires of the same current 1 meter apart and increased their current until the resulting force was exactly 2×10−7 N/m. They called this current 1 ampere (or 1 A), and were thus able to calculate µ0 as exactly 4π×10−7 H⋅m−1.

Example Equations

  • B = µ0I/r
  • ΦB = ∫ B•dA
  • B•ds = µ0I Ampere's Law
  • Emf = - {N} /dt Faraday's Law of Induction
  • B = µ0H (in a vacuum)

Finally, µ0 is measured in units of henrys per meter [h/m]

As the Magnetic Constant

Discovering a value for µ0 opened doors for researchers of magnetism, electricity, and later electromagnetism. It is simply a critical value without which accurate measurements of magnetic behavior could not be made. Likewise, many valuable theorems could not be postulated nor proven without the vacuum permeability constant. We certainly wouldn’t have dams, electric motors, electric vehicles, or electric guitars (to name only a couple) without it, which are all magnetic applications. Nearly all magnetic field calculations involving electricity utilize the constant of vacuum permeability.

...thought this would be a much appreciated relic, but the next video is more fun.

Vacuum Permittivity

Having a better understanding of vacuum permeability does not help in understanding vacuum permittivity; however, they are related -- more on that later. Vacuum permittivity and its relationship to electricity is slightly difficult to understand without a basic understanding of how “electricity” as we know it today works. If you do understand these concepts already, I recommend skipping to the "Permittivity Defined" section. For anyone who doesn’t, this is a good opportunity to learn how electricity and all of its ubiquitous wonders work.

Electric pylon, as seen everywhere, anywhere: roadside to mountaintop.
Electric pylon, as seen everywhere, anywhere: roadside to mountaintop. | Source

A Brief Summary of Electricity

First, allow me to set the scene. It is believed that the first fire seen by human eyes and utilized by human hands -- springing mankind to a new era of technology -- was created by a lightning bolt illuminating tumultuous storm clouds with several million volts. Throughout ancient history, cultures have been bewildered by light blue shocks and jolts, and by the magical properties of static electricity (when you shuffle across the carpet in your socks and then touch the television screen). Some cultures may have even successfully utilized this technology (Parthia's Baghdad Battery, for example). With very little understanding of the subject, it took modern culture quite some time to get a handle on the behavior and dynamics of electricity. First came detailed studies of static electricity, then the observations of electricity in biology (such as nerve impulses), and finally the observed link between electricity and magnetism (also known as electromagnetism).

A simple LED circuit.
A simple LED circuit. | Source

While all of these findings are individually important, allow us to focus on the fundamental aspects of electricity. Thanks to dear, wise Benjamin Franklin, we currently use the convention of positive (+) and negative (-) charges to describe attractive and repulsive forces. Similar to the attractive forces present in multi-body systems due to gravity, a force acts between two charged particles. This force attracts particles of opposite charge and repels those of similar charge. Negative charges are electrons, and positive charges are protons or holes (missing electrons). For simplicity, we will focus on the behavior of electrons. To review, we know that electrons are repelled by negative charges and attracted by positive charges. Luckily for us, many metals allow their electrons to flow somewhat freely (these are conductors), therefore we can send a charge through metal such as a wire or plate.

Artistic rendition (no kidding) of Benjamin Franklin's famous kite experiment, which demonstrated electric conductance.
Artistic rendition (no kidding) of Benjamin Franklin's famous kite experiment, which demonstrated electric conductance. | Source

Now picture a battery with a positive (nub) side and a negative (flat or dip) side. Imagine connecting a piece of wire from the positive side to the negative side of the battery. Recalling the behavior of negatively charged particles, we can expect them to move away from the negative side of the battery, and that is exactly what happens! The voltage of the battery is really called its potential difference (9-volt battery has a 9 volt potential difference), and refers to the difference in charge between the two terminals of the battery. This means that a higher voltage battery has a higher difference of charge that increases the effect on a charged particle (as we will further investigate soon). The repulsive charge on the particle combined with the opposite, attractive charge, causes it to move through the conductor in the direction of attraction. This direction or field of movement is called the electric field, and the combined motion of all charged particles in an electric field is called the current.

Being able to visualize the behavior of charged particles is half of the battle in understanding electricity - hope this video helps!

Permittivity Defined

Understanding the fundamentals of electricity, let us jump right in to permittivity. Similar to permeability, permittivity concerns the behavior of the material -- specifically, the dialectric medium -- when exposed to an electric field. A dialectric medium is one that can orient its charges in the presence of an electric field. When the medium experiences an electric field, its molecules polarize (or orient themselves) according to the direction of the field; this causes an internal electric field and also affects the ambient electric field. The measure of work required for polarization and resistance experienced during the orientation process is termed permittivity, and it occurs, to some extent, in all materials. If we consider a vacuum, we can imagine that there is no resistance to polarization and no effect on the electric field; thus we use vacuum permittivity as a reference to the permittivity of other materials. Permittivity is generally represented by the greek letter ε, while vacuum permittivity, a reference to all permittivity measurements, is represented as ε0.

Charles-Augustine de Coulomb: smug life.
Charles-Augustine de Coulomb: smug life. | Source


In a similar fashion to the derivation of vacuum permeability, scientists observed the existence of a constant of proportionality in experiments of electricity. The law of attraction in electricity, known alternatively as Coulomb's law, mimics Newton’s law of gravitational attraction but uses analogous electricity-related values, a similarity that was motivated, rather than discovered.

Newton's law of universal gravitation shows the relation between the masses of the bodies, the distance between them, and the proportional force that they create (either attraction or repulsion). The proportion governing this force is, of course, G.
Newton's law of universal gravitation shows the relation between the masses of the bodies, the distance between them, and the proportional force that they create (either attraction or repulsion). The proportion governing this force is, of course, G.
Look familiar? Coulomb's law shows the relation between the charges of the particles, the distance between them, and the proportional force that they create (either attraction or repulsion). The proportion governing this force is vacuum permittivity.
Look familiar? Coulomb's law shows the relation between the charges of the particles, the distance between them, and the proportional force that they create (either attraction or repulsion). The proportion governing this force is vacuum permittivity.

Again, scientists wanted the constant of proportionality to help in their calculations as opposed to hinder them. Thus, based on Maxwell’s equations, they defined the constant of vacuum permittivity to be equal to exactly related to the constant of vacuum permeability and the speed of light in a vacuum.

Where c0 is the speed of light in a vacuum. Ultimately, the value of ε0 was invented out of convenience, but the existence of its value holds great importance in applications of electricity and magnetism.


Example Equations

  • Φ = qi/ε0 Gauss' Law for Electric Flux
  • ∫ E•dA = q/ε0 Gauss' Law for Electric Fields
  • FC = (q1q2)/[(4πε0)(r2)] Coulomb's Law
  • E = Q/[(4πε0)(r2)] Derivation of Electric Field
  • V = Q/[(4πε0)(r)] Electric Potential (Voltage)

As the Electric Constant

Like the discovery and definition of vacuum permeability, analysis of the constant of vacuum permittivity shows that it is hugely useful and oft used in calculations of electricity and particle motion. Electricity is an essential part of our understanding of the universe. In our times, we certainly take advantage of its availability in large scale, but we can easily forget to appreciate its presence in the natural world. Just remember that our nerves function through electric impulse and everything we do, sense, remember, and feel is governed by electricity. In all aspects of electricity, ε0 is crucial to our understanding of underlying electric effects on and in materials.


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    • sehrm profile imageAUTHOR


      6 years ago from Los Angeles

      Oops! MsLizzy my comment was somehow truncated. I wanted to add that the trunk idea is really interesting, and I furthermore implore you to consider that the trunk may be empty, yet full of air. At the same time, all matter is about 99% empty due to the gap between subatomic particles, so no matter what you fill the trunk with, it will always be around 99% empty - interesting, no? Hope I've answered your questions. I look forward to reading more of your hubs!

    • sehrm profile imageAUTHOR


      6 years ago from Los Angeles

      Thanks MsLizzy! To answer your question, a short circuit occurs when a circuit element experiences some type of malfunction which causes it to have no resistance and thus an instantaneous, infinite current spike while maintaining variable voltage. This is a significant problem for elements connected in parallel, such as with American power outlets, and can cause damage to outputs. Connecting a battery from the positive to negative terminal would indeed cause damage to a conducting wire, as it does maintain some resistance resulting in heat which could cause it to overheat (don't try this at home).

    • sehrm profile imageAUTHOR


      6 years ago from Los Angeles

      Thanks for the comment Oztinato -- do you mean that you will be studying this in an upcoming course?

    • sehrm profile imageAUTHOR


      6 years ago from Los Angeles

      Thank you WiccanSage. It is quite difficult to explain these subjects, so I'm happy you think it's fascinating!

    • profile image


      6 years ago

      I will be studying this for several weeks. thanks.

    • DzyMsLizzy profile image

      Liz Elias 

      6 years ago from Oakley, CA

      A very well deserved HOTD. Nevermind that I understood little of it, for I don't know higher mathematics: (I'm lucky to keep my checkbook balanced)!

      I understand electricity and magnetism at probably an 8th-grade level at best.

      Which brings me to inquire, of your example of "connecting the positive end of the battery to the negative end..." (assuming no intervening appliance or switch) would this not merely serve to short-circuit the battery and discharge it, or worse, cause it to explode, in the case of some types of re-chargeable batteries?

      It was interesting, even if I only understood on a superficial level. Your coffee example reminded me of the old story about how much a large trunk could hold, and whether or not it was full after the addition of as many bowling balls as would fit. No, not full, because softballs would fit in between, and so on, down to golf balls, then sand, and finally, water before it was as full as its capacity.

      Voted up and interesting.

    • WiccanSage profile image

      Mackenzie Sage Wright 

      6 years ago

      Wow that is fascinating. My little mind has to stretch to grasp it but you explain things very well. Great hub.


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