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Why electric field not coming out of a conductor?

With the knowledge of High school physics, I assumed that I have learned following regarding Electric Current.

Current is moving charge and defined as amount of charge per second across the cross-section of a conductor.My old concept of current velocity and direction was current moving opposite to the movement of electron and since current speed is the speed of light,electron moving in the opposite direction with the velocity of light. This misconception was cleared by the fact the electron moving only with Drift velocity which is very much smaller than velocity of current which is actually the propagation of electro-magnetic field at the speed of light.

Now my question are,

1. Is it Electric field or Electro-Magnetic field which is giving current energy? If is it either what is it for DC current. Since DC has no changing field, EM field cannot be generated.

2. My concept is that : Current flow due to potential difference between the ends.
Why the field is following the path inside the conductor even if it is twisted, and not coming out of the conductor and jumps from the point of higher potential to the point of lower potential through shortest path. Why current induced Field is confined inside a conductor but not coming out of it.
What was the concept of wireless electricity in the mind of Nikola Tesla.

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Why the field is following the path inside the conductor even if it is twisted, and not comming out of the conductor and jumps from the point of higher potential to the point of lower potention through shortest path.

The electric field is present outside the conductor. In electrostatics or in quasi-static situations (when there is no substantial changing magnetic field in the system), the electric field is a conservative field, which means that if we take $$\int_a^b\vec{E}\cdot d\vec{\ell}$$ along any path between two points $a$ and $b$ we will get the same result. And that means any path, through a conductor or through the air around it, or by way of Albuquerque.

However no current flows outside the conductor, because there is no mobile charge there to move in response to the field.

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