Can we get induction without a transformer core? We usually see induction produced by two coils wound around a transformer core:
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Can you please specify if there is induction if we eliminate the core? Do we have to arrange the coils one inside the other? If the core can  increase induction, to what extent can we increase it?
What happens if, in the above transformer, we connect an appliance to both windings, do they both get half power or only the appliance connected to the second winding will work?
Also, can we get induction with straight wires?
 A: To answer the questions


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*Yes, there is still induction even if you remove the core of the transformer.

*No, you do not need to arrange the coils one inside the other, although this will increase the amount of current you induce (given that the two windings are electrically isolated from one another)

*We can increase the induction as determined by a parameter of the core known as the magnetic permeability $\mu$. This will depend on the type of metal you use as your core, and is quite large for iron.

*Connecting the transformer to multiple appliance from either winding will not do anything. In this case, there is no circuit. The intended use of a transformer is to take high voltage AC on the primary and induce current into the secondary winding, usually for the purpose of stepping down the voltage to something reasonable for consumer use.

*Can you get induction with straight wires? This is an interesting question, and the answer is yes. It is most evident with the differential form of Faraday's law: the presence of  a changing magnetic field will produce a curling electric field in the vicinity of the wire, inducing a current.


The primary coil generates an oscillating magnetic field which couples to the secondary coil. In fact, in an ideal world this oscillating EM field will propagate through all of space at the speed of light, and will couple to any coil that is any distance away. However, the effects of this field will be negligible as you move too far away, and in the case of an everyday transformer, it is necessary to incorporate the iron core to facilitate proper inductive coupling between the primary and secondary windings.
You can think about why this is the case with the integral form of Faraday's law of Induction, $$\mathscr{E_{2,1}} = - \frac{d\Phi_{2,1}}{dt}.$$
Suppose the secondary is a loop of wire that encloses a total area A, and that the magnetic field generated by loop 1 passing through loop 2 is $B(t)$ at time $t$ and $B(t+\Delta t)$ at time $t+\Delta t$.
What our good friend Faraday says is that in a short time $\Delta t$, the voltage induced in the 2nd winding is proportional to $A\frac{B(t+\Delta t)- B(t)}{\Delta t}$. It is intuitive then, that the strength of the magnetic flux passing through the loop is important, as having higher peak flux implies that the change in flux will be greater as well. 
The amount of current you will induce into the secondary winding will be significantly greater with the iron core, as the iron core traps the magnetic flux generated by the primary into a region of space that passes through the secondary. This is why it is so important to have the iron core in a transformer. 
A: The core of a transformer is normally made out of some sort of steel, which is the cheapest material, with a high magnetic permeability. 
If the core is removed, there will still be induction, although its flux is not guided through the air as it would do in the transformer and the field strength is much lower. The secondary side will not feel any voltage nor current.
Although if the wires on the primary and secondary side are of the same length, only electromagnetic radiation will be picked up, which is very minimal.
