A potentiometer wire AB having length $$L$$ and resistance $$12r$$ is joined to a cell D of emf $$\varepsilon$$ and internal resistance $$r$$. A cell C having EMF $$\varepsilon/2$$ and internal resistance $$3r$$ is connected. The length AJ, at which the galvanometer, as shown in the figure, shows no deflection is:

A

$$\frac{5}{12}L$$

B

$$\frac{11}{24}L$$

C

$$\frac{11}{12}L$$

D

$$\frac{13}{24}L$$