# Theoretical questions regarding the electric field

I found some theoretical questions in an old test of my school, and would like some help with the ones I cant answer:

1. It is said that the electric field is an intermediary of the electric interaction. What does this mean?

2. Why the participation of this intermediate is necessary and can not establish direct interaction between charges ( as supposed by Coulomb ) ?

3. Electrostatic field lines are open. How can you establish a relation between this and the fact that electric force is conservative?

4. If a charge is placed inside a non-uniform electrostatic field, will it move alongside the field lines? (I think this is true, because the electric field is parallel to the electric force which is what causes the movement state variation, but I'm not sure...)

1) This means that interaction between charged entities or matter takes place by the mediation of some force. According to standard particle theory, virtual photons are the mediators of electromagnetic interaction. Electric field is a certain region around a charge distribution, where it could influence a force on another charge. This means, the existence of some peculiar property of the two charges, called the electric field causes them to interact with each other. If you place a neutral material (of course, we are not assuming any induction), there will be no force on the neutral object, assuming the charge is very small.

2) Something is necessary to mediate something. Charges are just also a kind of matter. They are the sources of electromagnetic phenomena. Two charges are independent of each other. I mean, if you take two charges, their individual charges are not influenced by the other. But the charges act as a source o something by which they interact. This is called an electromagnetic field. A charge cannot, without the aid of this field, detect the presence of another charge. In other way, the electromagnetic field is the consciousness of a charge. This consciousness between two charged particles that are in the vicinity of both are interlinked by some mediator called the electric field of the individual particle. It works like a proximity sensor. The sensor is not conscious by itself. But it can be made to detect the presence of something by introducing something that is able to link the two.

3) Electrostatic fields are open. This means, in the absence of any external field, they travel out to infinity without being distorted. This shows that the electrostatic field shoes a divergent property. The field lines are not closed, unlike in a magnetic field, which means if you place a unit test charge $Q$ in the vicinity of a static positive charge $q$, it experiences a force given by Coulomb's law: $\vec{F}=Q\vec{E}$, where $\vec{E}$ is the electric field due to the charge $q$. This force accelerates the charge $Q$ and let this force displaces $Q$ from $a$ to $b$. The the work done is:

$$W=\int_{a}^{b} \vec{F}.d\vec{r}=Q\int_{a}^{b} \vec{E}.d\vec{r}=KQ\int_{a}^{b}\frac{\hat{r}}{r^2}.d\vec{r}=-\left[\frac{KQ}{r} \right]_{a}^{b}=-KQ\left(\frac{1}{r(b)}-\frac{1}{r(a)} \right)$$

If you put $r(a)=r(b)$, i.e., you come back from $b$ to$a$, then $W_{closed}=0$, which means the work done around a closed path by the electrostatic field is zero, which means the electrostatic field is a conservative field and the work done is path independent. The integral $W$ is called the potential difference between two points $a$ and $b$. This means, for a charge to be displaced from one point to another by the cat of an electrostatic field, there exists a potential difference between the two points.

This can be proved in terms of vectors also. Electrostatic field is irrotational. This means it's curl vanishes at every point. i.e.,

$$\nabla\times\vec{E}=0$$

Now according to Stoke's integral theorem for curl,

$$\nabla\times\vec{E}=\oint_{l} \vec{E}.d\vec{l}=0$$

which means the electric field is conservative in nature.

4) Now we have a non-uniform electrostatic field. In that case, the force acting on the charged particle is given by drawing a tangent to the field line at that point.

1) The electric field is not fundamental to the description of electrodynamics with point charges, one can take the point of view that electric charges simply interact at a distance with a force law proportional to the value of the electric field.

2) This is a bit of a loaded question in that any answer can normally be refuted, but the idea they're trying to get at is the idea of light, electromagnetic waves, or photons. The best way I'd answer this is the presence of radiation due to accelerating charges, since getting away without a field basically implies "spooky action at a distance," so Larmor radiation doesn't really make sense. But generally there will probably always be a way to construct the phenomenology of E&M without electric fields, so I don't love this question.

3) Open field lines mean the field has no curl. A field with no curl is the divergence of a scalar field. The divergence of a scalar potential is a conservative force. I'm not sure if there's something deeper than this...

4) Either parallel or anti-parallel depending on the sign of the charge, assuming they are not trying to "trick you" into forgetting states where the charge has initial velocity...