Running wire through header

Wife wants a light installed above the pass-through/breakfast bar. This pass-through (PT) is a 4x6 rectangular hole cut through the wall separating the kitchen and our living room. It's a load-bearing wall, so a header was installed (2 2x10s sandwiching 1/2 plywood). Header sits on a jack stud at each end which is adjacent to a sole to top plate stud ("king stud"?).

Anyway, the plan is for 3-way wiring to be run to this light. The supply line will have to come UP in a narrow chase created by the king/jack studs holding up this header and king/jack studs 5 inches away holding up a doorway header. All of this is interior framing. I was planning on drilling a 3/4' hole through the very top of the jack stud so that the wiring would come out at the bottom of and would be stapled to the header. A jamb will provide a chase for this wiring, and all will be encased in trim. At least, that's the plan.

Is there code that says I can't drill a hole through the king and jack studs for wiring? Or can you come down from the top of and go through a header (i.e. drill 3/4" through center of header)? Inspectors are due out tomorrow.

Thanks.

This is a building code question not an electrical code question.

personally I would say no

Inspection done?

I am curious. Did you drill the 3/4 hole? I suspect it was okay but would like to know for my own future reference.

Would she be satisfied with one of those stick up battery powered LCD closet lights?

Maybe a little late, but code requires that any wire that is not 1 3/8" from the edge of a framing member has to be protected by metal, ie a plate or metal conduit. No, you cannot run a wire in the "chase" made by the trim.