In the given figure the capacitor circuit continues to infinity. The battery connected has e.m.f. equal to V. The potential difference between points $$1$$ and $$2$$ is $$V2$$, that between points $$3$$ and $$4$$ is $$V4$$ and so on; i.e., the potential difference becomes $$12$$ after every step of the ladder. Find the ratio $$C1C2$$.

Question

In the given figure the capacitor circuit continues to infinity. The battery connected has e.m.f. equal to V. The potential difference between points $$1$$ and $$2$$ is $$V2$$, that between points $$3$$ and $$4$$ is $$V4$$ and so on; i.e., the potential difference becomes $$12$$ after every step of the ladder. Find the ratio $$C1C2$$.

Infinity Learn · Accepted Answer

Let the equivalent capacitance be $$C0$$. Even if we remove one ladder, the equivalent capacitance will remain $$C0$$.The potential drop across $$C1$$ in figure shown is $$V2$$.∴ $$q0C1=V2$$⇒$$q0=C1V2$$                     …(i)Also$$ $$q0=C0V$$                                           …$$(ii)∴ $$C12=C0$$⇒$$C1=2C0$$                       …$$(iii)From$$ the given circuit, $$C0=C0+C2C1C0+C2+C1$$⇒ $$C02+C2C0$$−$$C1C2=0Which$$ gives, $$C0=$$−$$C2$$±$$C22+4C1C22Only$$ positive sign is acceptable.∴ $$C0=$$−$$C2+C22+4C1C22$$∴ $$C12+C22=C22+4C1C22$$ ∵$$C12=C0C1+C22=C22+4C1C2C12=2C1C2$$⇒$$C1=2C2$$

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!