I’ve come across a lot of physics problems which ask of a current carrying wire has an electric field around it or will a current carrying conductor A induce charge on an adjacent grounded conductor B, and everywhere the explanation to the question is that current carrying conductors are electrically neutral. It’s stated like a universal truth everywhere and I can’t understand why does a conductor need to be neutral to carry current? Are there instances of charged conductors carrying currents? ( I understand that bit about electrons of the metal and the positive kernel cancelling each other’s charges. My question is, if I dump some charge on a current carrying wire, will it still carry current?) I’m in high school, so a simple explanation will be appreciated.
Why do current carrying conductors need to be uncharged?
They don't. Consider the simple case of a cell (battery) connected, via two conductors, to a resistor.
Though there is a current through the conductors, they are not electrically neutral; the conductor connected to the positive (negative) terminal of the cell has net positive (negative) charge.
As William Beaty explains here, this is necessarily the case since the energy flow from the cell to the resistor implies the existence of both an electric and magnetic field along the length of the conductor such that there is a non-zero Poynting field.
From the linked Wikipedia article:
Note the dashed electric field lines between the conductors.
Current-carrying conductors do not have to be uncharged, but any charge on them is likely to flow elsewhere when connected to another conductor. If they are charged, it does not affect their ability to conduct a current.
A power supply is an electric circuit made up of conductors. If you connect a charged conductor to another conductor, such as a power supply, the charge will be shared between the two conductors in proportion to their capacitances. A power supply usually has a very high capacitance, far larger than the capacitance of a wire. So most of the charge on a conducting wire would flow into the power supply circuit when they are connected.
Quite often electric circuits (especially power supplies) are "earthed" at some point, so the excess charge is then shared with the Earth, which has (essentially) "infinite" capacitance. Very little charge is likely to remain on the charged conductor.
One situation in which the conductor could retain its charge when connected to a power supply is when it is one plate of a parallel-plate capacitor. Provided that the other plate remains isolated and charged, almost all of the charge would remain on the current-carrying plate. It's current-carrying ability is not affected because a steady current does not change the amount of charge on the capacitor plate.
You are correct. If you have a closed circuit with a current and then add a charge to the conductor by some other means (not from the same battery that produces the current, but from some Van de Graaff generator or something similar), then indeed you would end up with a charged conductor carrying a current.
The meaning of the statement you see everywhere is that this charge cannot be added by using the same battery that produces the current, because the difference of the potentials (voltage) on the ideal conductor is zero. In other words, any charge on a conductor would immediately "run away" as a curent to compensate the opposite charge on the other end. Unless the entire circuit is charged by a separate source.