What does Field theory concern?

Question

I wounder to know what is the basic idea of field theory? Field theory wants to concern what aspects, and formulate what problems in nature?

Answer 1

A field is just a thing that attaches a value to each point in space. That value could be one-dimensional, in the case of a scalar field, or multi-dimensional, a vector field and you can also have other types of fields.

Field theory is very useful in analyzing forces. Take for example the forces due to gravity.

We know that two objects $1$ and $2$ with mass $m_1$ and $m_2$ respectively will feel a force $F$ between of magnitude:

$$F=G \frac{m_1m_2}{r^2}$$

Where $r$ is the separation between them.

Now we could say that object $1$ produces something that makes object $2$ feel a force. This something is what we call a field. In our example we can say that object one produces a field $E_1$ such that:

$$E_1=G \frac{m_1}{r^2}$$

Now $E$ is really a vector quantity but I think you get the idea.

Now we can say that object $2$ will feel a force when it is inside the field $E_1$ and the magnitude of that force is $m_2 \cdot E_1$.

This may seems like a triviality, but it is really rather useful. For example, when doing any sort of analysis of projectiles in the Earth, we can work out what downwards force an objects feels by using the expression $F=m \cdot g$, where $g$ is the value of the gravitational field on the surface of the Earth and as you may know is $9.81 \mathrm{ms^{-2}}$. This is much more simple than trying to solve the general formula, where we would need to know the mass of the Earth, the radius of the Earth and the value of $G$.

I hope this have you some insight.