# How does an electric field apply force to charged particles?

Im curious as how exactly a Electric field physically applies a force to positive and negative charges within its field. What interact action takes place? Does a charge simply “touch” the field and experiences a force at the point of contact? I am having a hard time conceptually understanding this and would really appreciate any help in visualizing this concept..cheers!

Edit: I understand we can not answer “why” questions with science but here I would just like to know “how” the field applies such a force. Whats the mechanism? Whats the interaction? Etc.

• Umm, through the creation of a magnetic field. Oct 21, 2022 at 0:48
– Babu
Oct 21, 2022 at 8:14
• By definition an electric field is a force per unit charge. What you are asking for is a tautology: How does a force (the field) apply a force? I don't see a good answer coming your way... ;) Oct 25, 2022 at 20:20
• I think that any satisfactory answer to this question would constitute philosophy, not physics. Oct 25, 2022 at 23:02
• @kricheli I got a -1 trying to explain that. For me, it's fine Oct 27, 2022 at 9:15

At the "intro to Electricity and Magnetism" level, the ability to support an electric field is a fundamental property of space, and the fields are governed by Maxwell's Equations. It is more or less axiomatic that the electric field is defined by the force it exerts on a charge.

At a graduate physics level, your question is more precisely answered by Quantum Field Theory, but I doubt that this is what you are looking for.

• @Jake My advice: Maxwell didn’t need quantum field theory to understand classical electromagnetism, and neither should you. Appreciate the beauty of classical field theory on its own. QFT isn’t going to help you do any of the problems in Jackson, for example. Oct 20, 2022 at 15:16
• @Jake One way to think of the electric field (in electrostatics only!) is that the field doesn't actually exist - only the interaction between charged particles at a distance - and the field is just a way to draw what the interaction would be at different locations Oct 20, 2022 at 21:30
• @Jake I just think of the field as “pushing” or “pulling” the charged particle. There isn’t a “mechanism” by which it happens. It is a direct interaction, at a point, between the charged particle and the field. Oct 20, 2022 at 22:09
• @Jake From my point of view, yes. Oct 20, 2022 at 22:46
• @Jake As an interesting point, "a hand pushing an object or a bat hitting a ball" are both interactions of an EM field. At the very basic level, particles aren't physically "hitting" each other, they are pushing each other apart via their generated EM fields. Oct 21, 2022 at 2:49

Note: this answer is a mathematical explanation, through a strictly classical lens, aimed at a presumably pre-high-school audience.

The electric force $$F$$ acting on a point particle with charge $$q$$ is simply defined as $$F = q \cdot E$$ with electric field $$E$$. In other words: wherever there is an electric field, there is a force field acting on the charged particle. These fields exist everywhere in space, and they take on different strengths at different points depending on what is generating the electric field. The way the field exerts its influence on the particle is through the charge: notice that if the particle had zero charge $$q = 0$$, then the force would be zero as well due to multiplication by zero. As such, the electric field can only exert itself on charged things.

To answer your question: there is no location where the particles "touch/interact" with the field, because they are always touching the field, because the fields exist everywhere. This "touching" is defined by the multiplication in this equation, and all other explanation is just an analogy to try and translate this equation into words. Which is to say: there are no words to explain it fully, you must learn the math.

Physics comes from math. Physicists speak to the public using words and analogy, but these concepts are only actually defined by equations, and all fancy language like "touching the field" (as you say) are a communication tool we've invented.

The (electric) field sets a landscape of potential energy $$e\, \phi(r)$$. The particle then moves in a corresponding way according to Newton's laws.

• Would you be able to elaborate a little further?
– Jake
Oct 20, 2022 at 13:26
• @Jake If I could elaborate for John, the analogy of a "landscape" is using our intuition for how hills and valleys influence how a ball might roll . A ball at the top of a hill will roll down it, accelerating at some constant rate. Analogously, a charged particle between two capacitor plates will travel from one plate to the other at a constant acceleration.
– Roy
Oct 21, 2022 at 10:04
• This answer is tautological, or doesn't answer the "how" (which, to be fair, I wouldn't know how to^^). Oct 25, 2022 at 17:40

There is a lot of depth to answering this fully, so I encourage you to look for a more detailed answer, but briefly:

The “electric field” is an entity whose energy is contained in virtual photons. The objects which are pushing on each other via the field are doing so by exchanging many virtual photons with each other. These virtual photons carry any momentum and energy which might be transferred between the objects.

• very nice ... I have been scratching my head about how to explain QFT at freshman level and this is quite elegant ...and just as @Ghoster says, it won't help anyone do any of the problems in Jackson! Oct 20, 2022 at 15:27
• By definition the virtual photons only pass force not energy. If energy is transmitted then that is a real photon! Oct 20, 2022 at 17:58
• @PhysicsDave I am far from a QFT expert, but my understanding is that a sharp distinction between virtual and real photons would not transfer easily to a classical picture. Virtual photons are very strange beasts which do basically whatever is required to bring our classical field into reality. That is enough of a definition for this situation I think. Oct 21, 2022 at 17:19
• I'm no QFT expert either, but aren't virtual particles only a visualization/intuition attached to terms in a perturbation series? Offering them as an explanation for a physical "why?" seems a bit unsatisfactory. Oct 25, 2022 at 20:13
• @kricheli It seemed to me that the question was not about the mathematics of classical theory, but about the story we tell ourselves as we make use of the theory. Virtual photons are a central part of the stories I tell myself about EM phenomena. As Kevin also points out in his answer, the energy in the electric field is the key here. But where, in terms of modern physics, can we find that energy? Virtual photons are the answer which works best for me, in most situations. Oct 26, 2022 at 2:03

If I may add a slightly left-field point of view - the thing to keep in mind when thinking about physics (or science in general) is that we are not really dealing with Truth as in something absolutely proven. The scientific method doesn't work that way - it is a tool that allows you to falsify hypotheses, and as you refine your theories, you construct something that sort of outlines an underlying reality (or that is what we hope). When I want to tease my phsysics colleagues, I say that physics is the best lie we could come up with ;-)

So, to get back to your question - the short answer is that we don't really know. Our experience, through experimentation, suggests that there is something, which we call electric charge, which interacts in a characteristic way, and which is different from that other something called mass; to give us an intuition about how these things can influence each other at a distance, we have invented the concept of a field. The thing to remember is that all these concepts are probably nothing more than words that allow us to talk about our experiences and make accurate predictions about things.

Involving quantum mechanics doesn't really add to our understanding - QM is in fact famous (or notorious, if you like) for not giving us an understanding of things. It is a marvellous tool for making calculations about large assemblies of very small thing, I think most quantum physicists will tell you that we don't really understand why.

• Can you write down a name of a book that elaborates more about the idea you mentioned in the first paragraph? I am interested to read more about this way of thinking.
– tush
Oct 21, 2022 at 13:04
• @tush It is just my take on the scientific method - Karl Popper's 'The Logic of Scientific Discovery' may be the best source: or you could start with wikipedia: 'en.wikipedia.org/wiki/…' Oct 21, 2022 at 14:11

"The electric field has a direction and magnitude indicating the electric force that would exist on a unit-charge placed at that location, ignoring its own field."

The important thing is that the electric field is already defined using the force it creates. An electric field is like an open door. An open door doesn't do anything (you can walk through), but it provides the possibility of being shut and then doing something, but once it is shut it is no longer an "open door". A static electric field with no charge in it doesn't do anything, although it implies the possibility that if you put a charge it would do something. (But once the charge is actually there then you don't need the electric field anymore.)

Its entirely possible to write down the whole theory of electrostatics without ever brining in the electric field. We only ever see what actually happened, not the "what if", so all we ever see is force. Electric field is a tool for talking about the forces we think we would have seen if only something were different.

So the question "how does the electric field produce a force" is resolved. The reason we decided there was an electric field at that location is because we believe that a charge at that location would feel a force. Given that we believe that a charge at that location would feel a force, then we believe it a charge at that location will have a force produced on it.

## What does asking "why?" really mean?

At the base of math are axioms which build theorems. At the base of science, especially physics, we have first-principles which are supported with experiments.

Asking why is asking "how does (something closer to first principles) lead to (something farther than first principles)". For example: "How does a billiard-ball model of a gas lead to an airfoil's lift?". You could also ask "How does quantum mechanics lead to the billiard-ball model?" to dig further down.

## Energy, not force, is a first principle

In classical physics we are taught that, for example, springs exert a force. We integrate this force over distance to get the total energy.

However, quantum mechanics works with the Hamiltonian, or total energy. This is conserved; friction only appears in many-body systems where the energy gets transferred to a lot of particles in a chaotic way (heat).

Forces arise from energy gradients. In the time-dependent Schrödinger equation the velocity of a wave packet is proportional to the phase-gradients across said packet, see this interactive demo. All else equal, phases change faster in higher-potential areas. If a particle is on a slope, the phase will change at different rates across the packet, building up a gradient across the particle. Causing it to accelerate down the slope.

An energy-field, not a force-field

An electric field E corresponds to a field of energy density proportional to |E|^2. This is encoded explicitly in quantum field theory: it appears in part of the $$F^{\mu \nu}F_{\mu \nu}$$ term of the QED Lagrangian.

If you move a charged particle around, the integral of |E|^2 changes. It turns out that the gradient of the integral is simply the E-field (at the particles location but without the particle) multiplied by the particle's charge.

Two charged particles can exert a force on one another over a distance. This may be verified and measured experimentally. No one truly knows how or why this happens. Assuming this force as a fact, however, we can do calculations and predictions. And to aid those calculations and predictions, we can rephrase this in terms of a mathematical construct that a priori exists purely in our minds and in our calculations: the electric field.

If we use this electric field theory, then instead of particles affecting one another directly, a charged particle is associated with an electric field, and that electric field again will exert a force on any other charged particles. The electric field is manufactured in such a way that it is a perfect stand-in for particles affecting one another directly. It has to be, otherwise its predictions would disagree with experiment.

So an electric field can exert a force on charged particles because we invented the electric field precisely to assist in calculations involving charged particles that somehow exert forces on one another, and to be the mediator of these forces.

The operative definition of the electric field can come from the experimental measurements of the force $$\mathbf{F}$$ acting on a test charge of known (and small, to reduce the intrusivity of the test charge) intensity $$q$$.

(To get the definition of the electric field, you need to remove all the forces acting on the test charges that don't come from Electromagnetism, like the weight of the charge or charged object)

The electric field $$\mathbf{e}(\mathbf{r})$$ can be defined as the ratio of the force $$\mathbf{F}$$ acting on an electric charge and the electric charge $$q$$ itself, positioned in the point $$\mathbf{r}$$.

As you move the test charge in space, you measure different values of $$\mathbf{F}(\mathbf{r})$$, so that the definition of the electric field reads

$$\mathbf{e} (\mathbf{r}) := \dfrac{\mathbf{F}(\mathbf{r})}{q}$$.

The measured force $$\mathbf{F}(\mathbf{r})$$ acting on the test charge depends on all the electromagnetic phenomena influencing the domain, and it's the resultant of their interaction with the test charge.

• I understand how we calculate and find the electric field, but Im more wondering “how” this entity of space enparts a force of charged particles? Is it more of a “reaction” so to speak?
– Jake
Oct 27, 2022 at 14:48

Even there is no Source there is always a field Imagine like this. We have a river. Now in the river we assign every point to wave like thing that moving equally probably in all directions more generally what physicist call 'spherical wave' a green function or a propagator . Now the field i am talking about is not Electric or magnetic field it is gauge field its directions is given by the propagation vector 'k' . Here comes the picture .

imagine we have uniformly distributed continum light source such as led in a big area of space. They are uniformly emitting light in all directions . Now imagine a charge particle or say a spherical ball that floating in field and those leds are propagating light in field medium now since the spherical ball floating in field it will also get displaced from its orginal position. But since he getting bombarded by waves from all directions its net force is zero . So it will not gonna displace. now a electric field can be regarded as a brighter light that the same light excited due to energy the energy of electric field is like more directed light in certain direction thats when distrub the field that make the ball moves in certain direction

Now the light i talk of about is called vertual photon. The bright light means a photon is created out of vacuum by creations opperator these are quantum mechanical things . The field i talk of is not only a scalar field there is also vector field these are called gauge field . That vector field has tendency to induce directions in certain region of space out of vacuum make charge particle follow it

Directed flow in physics language can be understood as liner combination of plane wave the fourier coefficient.

conclusion is the electric field distrub the uniform behavior of space by exciting(i will take away some water resulting all floating objects moves towards me or i will add more water at some point resulting near floating objects get further away)the field resulting all particle in the field follow certain direction

This is probably the least sophisticated answer you have gotten. But here goes:

Im curious as how exactly a Electric field physically applies a force to positive and negative charges within its field.

Physical interactions imply physical contact between objects. In that sense, the force the electric field exerts on a charge is not "physical", but rather what is sometimes referred to as "action at a distance".

Does a charge simply “touch” the field and experiences a force at the point of contact?

Again, "touch" implies physical "contact". In that sense, the charge does not "touch" the field, but rather interacts with the field. The concept of a field was needed to explain the hitherto lack of understanding of how forces can occur on bodies without physical interaction with other bodies.

It was up to Michael Faraday to propose the "physical existence" of lines of force associated with the existence of a field.

Hope this helps.