On both sides of a magnetic needle, two short magnets $A$ and $B$ are placed on the same horizontal line which is perpendicular to the magnetic meridian. The south poles of $A$ and $B$ are facing each other, which are $10\text{ cm}$ and $20\text{ cm}$ respectively from the magnetic needle. If the needle remains undeflected, the ratio of the magnetic moment of $A$ to that of $B$ is: