The question asks for the output waveform across a capacitor for a given input waveform, but the actual circuit diagram and the waveform/option figures are not visible in the provided prompt.

To determine $V_0(t)$ across a capacitor, we must know:

1. The circuit configuration

   • Is it an $RC$ low-pass circuit?
   • Is it a high-pass circuit?
   • Is there a diode/capacitor clamper or differentiator/integrator?

2. The exact input waveform $V_{in}(t)$

   • Square wave, triangular wave, pulse train, etc.

3. The option plots A, B, C, D

   • Since this is an MCQ based on waveform matching, the figures are essential.

Without these, the answer cannot be derived rigorously from first principles.

However, the stored correct answer is given as B. Since no diagram is available to independently verify, I cannot honestly claim a derived agreement from physics reasoning alone.

What would normally be done

For a capacitor output in an $RC$ circuit, we typically use:

$i_C = C\frac{dV_C}{dt}$

and in an $RC$ network,

$V_{in}(t) = V_R(t) + V_C(t)$

with

$V_R = iR = RC\frac{dV_C}{dt}$

so that

$RC\frac{dV_C}{dt} + V_C = V_{in}(t)$

Then we solve piecewise over each interval of the input waveform and compare the resulting shape with the options.

Conclusion

Because the required waveform figures are missing, the problem is incomplete in text form. I therefore cannot derive the correct option from the information provided alone.

Given only the stored answer, the option is B, but this is not independently verified.