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An isolated neutral cylinder has an equal number of positive (red) and negative (blue) charges.
The electric field lines from a test charge placed near the cylinder will pierce the cylinder.
The electrons (negative, blue, charges) will move in response to the electric field lines, going towards the test charge.  This leaves an excess of positive charge on the cylinder’s far side.
As time goes on, more and more electrons will move towards the test charge. The electrons close to the cylinder, and the left-behind positive charges on the far side, will begin to cancel the electric field of the test charge.  Electric field lines from the test charge will begin to terminate on the cylinder.  New field lines will originate on the far side of the cylinder.
The electrons will stop moving towards the test charge when enough have moved to completely null the electric field inside the cylinder.  Because the test charge is overall neutral, far away the field will be identical to the field from the test charge alone.