Do we need waves for fields? I was pondering about EM Waves and fields and felt that there is an inconsistency in the physical picture of EM waves that I have in my mind. For example let us consider a charge at rest . Now lets say we want to test the electric field due to this charge Q at a point P. So to do this we need to place another charge at that point and find if we can feel any force due to that charge. Now I think for Q to exert a force on test charge or an electric field to be established at that point there must be EM waves propagating from the charge Q to the test charge. However there are no accelerating charges which are the prerequisites for EM waves to be generated . So this brings in an inconsistency in the physical picture of EM waves I have in my mind. I believe that there is some misconception I am having and I am positively looking forward to replies to clear it out.
 A: 
Now I think for Q to exert a force on test charge or an electric field to be established at that point there must be EM waves propagating from the charge Q to the test charge.

A wave would imply a forces that changes in time, but the force does not do that.
You could argue that you are talking about very long periods, but the DC limit is electrostatics.
A: There is no need for a wave for a force to be exerted. The formula for the Lorentz force is $\mathbf{F} = q\mathbf{E} + q\mathbf{v}\times \mathbf{B}$. What this says is that even for a static electric field, such as that produced by a charge at rest, the test charge will feel a force $\mathbf{F} = q\mathbf{E}$, even if the field is not changing with time.
i would say your misconception lies in the statement "Now I think for Q to exert a force on test charge or an electric field to be established at that point there must be EM waves propagating from the charge Q to the test charge". Electromagnetism simply does not work like that. Waves will certainly cause a test charge to move, but they are not the only way. In fact, since waves are a combination of oscillating electric and magnetic fields, they will cause the test charge to move in a rather complicated way. A charge $Q$ at rest, however, produces a field $\mathbf{E} = \frac1{4\pi \epsilon_0}\frac{Q}{r^2}\hat{\mathbf{r}}$, and a test charge placed in such a field will move in a straight line, a quite different motion to that produced by a wave.
A: your confusion is easy to resolve. think of Maxwell's equations. now look at them again but assuming that there is no time dependence, so all time derivatives will equal to zero. you'll get Gauss's Law this way. it's for static charges. you put a charge, it'll put a force on another charge as you described. no waves involved.
the waves are periodic solutions of the same Maxwell equations, they appear when there's time varying component, such as moving charges
