Question 500: Electrostats & Capacitance

Question

Two parallel plate air capacitors of same capacity ' C ' are connected in series to a battery of emf ' $E$ '. Then one of the capacitors is completely filled with dielectric material of constant ' K '. The change in the effective capacity of the series combination is

Accepted Answer

Let's break down the problem step by step.
Two identical air capacitors, each of capacitance $$C$$, are connected in series. The effective capacitance in series is given by
$$\frac{1}{C_{	ext{series}}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C},$$
so
$$C_{	ext{series}} = \frac{C}{2}.$$
Now, one capacitor is completely filled with a dielectric of constant $$K$$. When a capacitor is fully filled with a dielectric, its capacitance increases by a factor of $$K$$, so its new capacitance becomes $$K C$$. The other capacitor remains at $$C$$.
For two capacitors in series (one with capacitance $$KC$$ and the other with $$C$$), the effective capacitance becomes:
$$C' = \frac{(KC) \cdot C}{KC + C} = \frac{K C^2}{C(K+1)} = \frac{KC}{K+1}.$$
The change in effective capacitance is the difference between the new effective capacitance and the original:
$    \Delta C = C' - \frac{C}{2} = \frac{KC}{K+1} - \frac{C}{2}.    $
To combine the terms, find a common denominator:
$    \Delta C = \frac{2KC - C(K+1)}{2(K+1)}    = \frac{2KC - CK - C}{2(K+1)}    = \frac{C(K-1)}{2(K+1)}.    $
Thus, the change in the effective capacitance is
$$\frac{C}{2}\left(\frac{K-1}{K+1}ight),$$
which corresponds to Option C.

Answer

$\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}+1}{\mathrm{~K}-1}\right]$

Answer

$\frac{2}{\mathrm{C}}\left[\frac{\mathrm{K}-1}{\mathrm{~K}+1}\right]$

Answer

$\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]$

Answer

$\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]^2$

Electrostats & Capacitance

Two parallel plate air capacitors of same capacity ' C ' are connected in series to a battery of emf ' E '. Then one of the capacitors is completely filled with dielectric material of constant ' K '. The change in the effective capacity of the series combination is

Choose the correct answer: