In a time varying magnetic field why is the current flowing through a conducting wire connecting two points of a closed conducting loop zero?
Try this.  Cut the loop in half with a horizontal conductor as you have drawn the picture.  Now cover the top half and draw your arrows showing the current in the bottom half.  Now cover the bottom half and draw the top half arrows.  Now uncover the whole thing.  What happens to the arrows on the horizontal conductor?.  Got it?