Final answer:
To cancel out the z-component of the electric field at the origin generated by point charges at (3m, -2m, -9m) and (6m, -5m, 1m), calculate the z-components of their fields at the origin and determine the necessary magnitude of a third charge at (0,0,3m) to produce an equal but opposite z-component.
Step-by-step explanation:
To determine the charge that would create an electric field at the origin with no net z component, one must consider the effects of the other two charges already placed in the space. The existing point charges of 1μC at (3m, -2m, -9m) and -3μC at (6m, -5m, 1m) create electric fields that contribute to the z-component of the electric field at the origin.
We want to place a third charge at (0,0,3m) such that it cancels out the z-component of the net electric field due to the other two charges at the origin. Knowing that the electric field produced by a point charge decreases with the square of the distance, we can use Coulomb's Law to calculate the necessary charge.
Firstly, compute the z-components of the electric fields from the two existing charges at the origin. Then, knowing the distance and direction from the third charge's position to the origin, find the magnitude of the required charge that would produce an electric field of equal magnitude but in the opposite direction to the sum of the z-components of the electric fields from the existing charges.
This is a complex calculation that involves vector components, the principle of superposition, and may require the use of calculus, depending on the precision required for the calculation.