The circuit diagram is not visible in the prompt, so the exact capacitor connections are missing. Without the actual arrangement of capacitors, the value of $C$ cannot be derived uniquely from the given information alone.

However, since a stored correct answer is provided, I can only make a consistency check against the options:

1. The required equivalent capacitance is $0.5\,\mu F$.
2. Depending on the network, values like $\dfrac{6}{5}$, $\dfrac{7}{11}$, $4$, and $\dfrac{7}{10}$ $\mu F$ could all be plausible for an unknown branch capacitor.
3. But without the circuit topology, there is no valid step-by-step reduction possible.

So, strictly speaking:

• The problem is incomplete as stated because the circuit figure is required.
• Hence the answer cannot be independently verified from first principles.

Given the stored correct answer, the marked option would be:

$\boxed{C=\frac{7}{11}\,\mu F}$

which corresponds to Option B.

If you upload or describe the capacitor network, I can solve it properly step-by-step and verify whether $\dfrac{7}{11}\,\mu F$ is indeed correct.