# Q: Does an electric field have mass? Does it take energy to move an electric field?

The original question was: An electric field stores energy.  Energy has mass if I understand E=mc2 correctly.

Now imagine a lone electron. It has an electric field. And therefore that field has mass presumably. If I apply a force to that electron, it will accelerate according to F=ma. My question relates to the m in F=ma.

The electric field must still exist even when the electron is moving. So therefore I am ALSO “moving” the electric field as well as the electron. So the m in F=ma must be made up of two parts, one of which is the mass component of the electric field and one of which just relates to the electron itself? Is that correct or am I confused. PS I appreciate it will be incredibly small and I also appreciate there may also be a magnetic field due to the changing electric field.

Physicist: You’re exactly right.  The electric field has mass (or, at the very least you could say that it has inertia and attracts things gravitationally), because it carries energy.  The energy density, K, of the electric field around a charge, q, is $K=\frac{q^2}{R^4}$ (ignoring all the physical constants for simplicity).  Near the charge (R=0) this equation doesn’t quite work, because the electron isn’t a point, but otherwise it holds up.

You can think of the energy in the field like a mess of Jello™ that’s thick near zero then thins out in all directions.  If you push the charge in the middle, the Jello™ will also move, but the movement will take the form of a jiggly wave that propagates outward.  That wave is where all the extra energy goes.

Electromagnetic energy.

Dropping the metaphor; pushing on a charge generates an electromagnetic (EM) wave.  So applying a force to something with a charge (like electrons) takes more energy than it should (based on the mass alone), because the act of pushing on it generates a spray of photons (which is light, which is EM waves).

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### 5 Responses to Q: Does an electric field have mass? Does it take energy to move an electric field?

1. Ethan says:

WOW!!!! so cool!!! Thanks for sharing your knowledge!!!

2. Aurthi says:

An electric field that can balance an electron of mass 3.2×10power-27 kg is———–

3. Daniel Kovacs says:

Actually there is a big misunderstanding in the physicist’s answer which seems to be very common for some reason even among experts. He says that the electric field has mass because it has energy stored in it. This is simply not true. Not all forms of energy have an equivalent mass. For example, the photon has no rest mass but carries energy, E. Just by using Einstein’s formula and saying that m=E/c^2 is a kind of mass that we can attribute to the photon actually brings about a confusion over the role of the concept called mass in physics. Because although this formula is true, it means nothing in this case. What’s more, this kind of mass is not even unique because it is transformed if we change the frame of reference. And since it is transformed in the same way as energy, it is basically the same as the energy and coining another name for it does not lead to another physical quantity.

We must understand that mass is not a prime concept in relativity, only another form of energy (which is called rest energy) and just a part of the total energy of a particle or macroscopic object. While in nonrelativistic physics mass, energy and momentum are all conserved during the physical processes, it turns out that in relativistic physics mass is not conserved at all. It composes only a part of the total energy of an object that must be taken into account when we want to keep track of the total energy balance. Only energy and momentum are conserved so these two remain to be prime concepts but mass does not.

So, based on this, what can we say about the mass-energy equivalence? That it is true as long as we can give a meaning to mass independently of this formula as well, so that the formula creates equivalence between two quantities existing in their own right. For example, we can measure the mass of an electron independently of its electric field so deriving its mass from its electric field does make sense. But saying that the electromagnetic field in general has mass because it has energy makes no sense at all. EM field has energy, linear momentum and angular momentum because these are all conserved quantities and so prime concepts. But mass is something which does not necessarily have any meaning in a situation. So I would say that we have to be cautious with using it too frequently.

As for the other statement (that EM filed attracts things gravitationally), it is true. But not because it is its mass that attracts things gravitationally but because the source of gravity is not mass alone but the energy-momentum tensor. This has 16 components of which only one is the mass. The rest are energy and momentum density currents and the stress tensor.

Unfortunately (or not unfortunately) electromagnetism is intrinsically relativistic, so its deeper understanding can not lack the understanding of special or even general relativity as well.

4. Doug Robinson says:

The similarity between Newton’s second law and Einstein’s mass-energy equation cannot be denied. F=ma and E =mc^2 which show that both force and energy are equal to the product of mass and acceleration, albeit when the acceleration term is c^2
That this is true is demonstrated in the confined electrostatic fields of a fully charged capacitor. That field is measured in Newton’s per coulomb, i.e force and charge which provide a means of efficient storage of energy, the normal application of capacitors.
The point made here is that contrary to the notion that it does not make sense to associate electric fields with mass, rearranging Newton’s F=ma shows not just E/c^2 = mass, but also F/c^2 =mass. Thus the potentially phenomenal forces that can exist in the electrostatic field between the surfaces of a capacitor, do indeed equate to inertial mass. Capacitors are a large scale demonstration of the binding energy between protons and electrons, and the mass that is associated with the forces of attraction inside atoms, electric forces estimated to apply down to 10^-13 cm according to an analysis by Richard Feynman, Vol 1. ch 12 Feynman lectures on Physics.