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Two questions:

First, how does the gate provide a positive charge by running a current through it? I really just don't see how this positive charge could be achieved, unless you had some sort of battery in the gate itself and attached a wire to donate some of the electrons flowing through to the cathode of some other battery. Is it possible to get electrons out of the gate by some other means?

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    $\begingroup$ perhaps this should be asked on Physics.SE $\endgroup$ – user15489 Apr 16 '15 at 9:18
  • $\begingroup$ Is there a significant $I_{GD}$ in FETs? In contrast to BJTs, the former aren't current-, but voltage-controlled. A conducting channel between source and drain is typically induced by applying a (positive) voltage between gate and source. $\endgroup$ – Klaus-Dieter Warzecha Apr 16 '15 at 9:29
  • $\begingroup$ @KlausWarzecha Applying a positive voltage on the gate means that electrons are moving into the gate, right? But wouldn't that make the gate more negative? $\endgroup$ – extremeaxe5 Apr 16 '15 at 10:30
  • $\begingroup$ Why wouldn't the electrons move to the positive voltage source? Instead of Physics SE I might suggest Electrical Engineering. But, you seem to have a number of concepts mixed up with each other, so a little more thought should be placed before re-asking the question. $\endgroup$ – Jon Custer Apr 16 '15 at 13:37
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The gate of a FET works much like the grid of a radio tube (valve). The gate is insulated from source and drain; the only current flow (if the insulator were perfect) is to charge the intrinsic gate capacitance. Even a MOSFET has some leakage, but it's pico- or femtoamperes. See this discussion on capacitance and leakage.

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It is really not charge injection, but charge separation. When a potential is applied to the gate with respect to the SD, the charges in the gate dielectric separate. This separation creates a layer of positive or negative charges just next to the SD channel.

In ideal cases there is no current through the gate.

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