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If you had taken any other orientation, the length of rod enclosed would have been more. ( Like if you had taken the rod to be passing from the center through the edge of the cube, the length enclosed would have been L/(root 2), hence the charge enclosed would have been more.)
Equivalently, you can see that the shortest distance between the center of the cube, and the 'outside' of the cube would have to be L/2. Hence minimum amount of the rod with one end at the center (and charge) which could be enclosed would be L/2 (Q/2).
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