# Is electric field a property of a charge or is it a spatial distribution of electrostatic force?

I am a bit confused about what gives rise to an electric field. I can look at it in two different ways as follows.

1. When two charges are separated by a distance, the electric field gives the distribution of magnitudes of electrostatic force at different sets of two different positions of the two charges within the field suggesting that the electric field exists only if the electrostatic force exists. And will not exist if there is only one charge present.

Or

1. As quantum theory of fields suggests, the electrostatic field is a property of a charge and thus exists even when there is only one charge present suggesting electric field to be independent of electrostatic force.

Which of the above two statements is correct? Is electric field linked to the charge or the electrostatic force?

• Your first statement doesn't suggest to me that the electric field only exists if the force exists. It only says that the distribution of the forces is given by the electric field.
– user112876
Commented Apr 28, 2021 at 11:52
• @Ezze In my first statement: The distribution of magnitude of force describes the electric field. The force is the discriber of the field (that is discribed). The discriber must exist to describe the discribed. Suggesting that electric field exist only when the electrostatic force exists. Commented Apr 28, 2021 at 11:57
• "If a tree falls in a forest and no one is around to hear it, does it make a sound?" If there's nothing to measure the force with, does the field even exist?
– user112876
Commented Apr 28, 2021 at 19:24

The classical definition of an electric field at a position $$x$$ is the force that a small test charge would feel if it was placed at $$x$$ (divided by the test charge).

More formally,

$$$$\vec{E}(\vec{x}) = \lim_{q\rightarrow 0} \frac{\vec{F}({\rm on\ }q{\rm\ at\ } \vec{x}\ {\rm\ due\ to\ other\ charges})}{q}$$$$

In particular, there is no problem defining the electric field due to a single point charge.

In quantum field theory, the electric field is an operator that is typically defined in terms of the electromagnetic tensor $$F_{\mu\nu}$$ $$$$E_i \equiv F_{0i} = \partial_0 A_i - \partial_i A_0$$$$ where $$A_\mu$$ is the vector potential. There need not be a source of the electric field in quantum field theory; vacuum fluctuations have physical effects such as the Casimir effect, or renormalization of the electron mass.

• What about the definition of electric field from the quantum theory of fields? If a test charge is necessary to describe the electrostatic field of another charge, it means the field is not a property of the charge but a discription of the distribution of electrostatic forces experienced by the two charges. Right? But the quantum theory of fields suggests that the electric field is a property of the charge. Commented Apr 28, 2021 at 12:12
• @Somanna I added a comment about qft. Commented Apr 28, 2021 at 12:17