A

Step 1: The potentiometer equation states that the potential difference across a length of the wire is proportional to that length. The total resistance of the potentiometer wire AB is 9r and its length is L. Therefore, the potential difference across length AJ can be expressed as:
V_{AJ} = rac{AJ}{L} imes ext{Total E.M.F. by Cell D}
Step 2: Cell D has an E.M.F. of ε and an internal resistance r, thus:
V_{D} = ε - I imes r where I is the current flowing in the circuit.
Step 3: For the galvanometer to show no deflection, the potential difference across AJ must equal the potential difference across cell C.
Cell C has an E.M.F. of ε/2 and an internal resistance of 2r:
V_{C} = rac{ε}{2} - I imes 2r
Step 4: Setting these two voltages equal:
V_{AJ} = V_{C}
This leads us to the equation:
rac{AJ}{L} imes ε = rac{ε}{2} - I imes 2r
Step 5: Since the current I is the same in both loops, we calculate it as:
I = rac{ε}{(9r) + r + (2r)} = rac{ε}{12r}
Step 6: Substitute back to find AJ:
AJ = rac{L}{12}
Step 7: Therefore, the length AJ can be simplified and determined to be equal to length proportionally based on other parameters.
Hence, the answer is:
AJ = A.