What happens when a potential difference is applied across an inductor? I am trying to get a picture of mechanism from start, just after potential difference is applied in detail? In resistor we can see that there are positive ions opposing and current flows at constant rate so energy dissipated in resistor is equal to that given by battery. But what is happening in conductor so that the induced EMF gets equal to E? What is the picture of the mechanism?
Essentially, if the current through a loop of wire is changing, the associated magnetic field threading the loop is changing and, by Faraday's Law, there is an associated induced emf.
Now, to your question:
But what is happening in conductor so that the induced emf gets equal to E ? What is the picture of the mechanism ?
Let's assume the wire making up the loop is an ideal conductor or effectively so. Now, recall that there can be no electric field inside an ideal conductor*.
With the voltage source E connected to the loop of wire, the only way for there to be no electric field in the conductor is for there to be an induced electric field that precisely cancels the field due to the voltage source.
Since the requirement for an induced emf is a time changing magnetic field and associated time changing current, the time rate of change of current is precisely the value required to induce an emf that cancels the applied voltage.
*there are conditions in which this is not true but these are beyond the scope of this answer.
A resistor and inductor both resist current flow, but the inductor opposes CHANGES in current while the resistor simply opposes the MAGNITUDE. The inductor opposes changes in current because changes in current cause a magnetic field (or back EMF) that pushes back against the current trying to flow.
Here is an Inductor analogy: Let's say you push a guy using a constant force. If he is a resistor, he immediately starts moving at a constant rate. And if you stop pushing him, he immediately stops. A big guy (large resistor) moves slowly and a small guy moves fast. Easy to understand, right? Now, lets say he is an inductor. The instant you push him he feels like a brick wall, but then he starts to move faster and faster and faster.... then you stop pushing him and then he continues to move but slower and slower and slower.... until he stops on his own. A big guy (large inductor) will be sluggish compared to a small guy. In this analogy, the force=Voltage and speed=Current. I suppose you can say that a resistor has no momentum properties while an inductor does. Hope this helps... now on to the math...
One of the best ways to analyze inductor circuits is to use the well known equation for an inductor:
V = L*di/dt
Rearranging, we get this: V/L = di/dt
What does this mean? It means that the rate of change of current through the inductor is constant and equal to the voltage applied divided by the inductor value. If we integrate the equation we get: i(t) = V/L * t
So, in time, the current through an inductor is a simple linear ramp (y = mx). This type of analysis is used mostly in Switch Mode Power Supplies (SMPS) where the inductor is used for DC/DC conversion (i.e. 5VDC to 3VDC).
Hope that helps. By the way, in this figure, the back EMF is equal to V because the voltage source V is an ideal source that dissipates any transient back EMF. But note that from my figure, when the switch opens up, the back EMF voltage will go to infinity because there is no ideal voltage source there to keep it from doing so. Why does it go to infinity? Because the change in current was infinite (the switch stopped the current instantly!). V = L * INFINITY!!! So, if you push an inductor guy and he is moving, you can't stop him instantaneously because he'll rip your arm off. (-: If you are superman (ideal) I guess you can do it. This is why inductive loads quite frequently produce large voltages that can blow stuff up. Fun stuff...
In your case R is the ESR of the battery (you may want that to be zero but then you leave the real world)
What is happening is that electric and magnetic fields are interacting. I don't think you can easily model that in terms of physical analogues.