A parallel plate capacitor \((C)\) is connected to a battery as shown in the figure. Consider two cases:
Case-I: Key \(k\) is kept closed and plates of the capacitor are moved apart using an insulating handle.
Case-II: Initially key \(k\) is closed for a long time and then opened. Now, the plates of the capacitor are moved apart using an insulating handle.
Identify the correct statements among the following:
(A) In Case-I, \(Q\) remains same but \(C\) changes.
(B) In Case-II, \(V\) remains same but \(C\) changes.
(C) In Case-I, \(V\) remains same and hence \(Q\) changes.
(D) In Case-II, \(Q\) remains same and hence \(V\) changes.
Choose the correct answer from the options given below:
Show Hint
For capacitor problems:
\[
\text{Battery Connected}
\Rightarrow
V=\text{constant}
\]
\[
\text{Battery Disconnected}
\Rightarrow
Q=\text{constant}
\]
This is the most important rule for variable-capacitance questions.
Concept:
For a parallel plate capacitor,
\[
C=\frac{\varepsilon_0 A}{d}
\]
When the plate separation \(d\) increases,
\[
C \downarrow
\]
Also,
\[
Q=CV
\]
The quantity that remains constant depends on whether the capacitor remains connected to the battery.
Step 1: Analyse Case-I (key closed).
Since the capacitor remains connected to the battery,
\[
V=\text{constant}
\]
As the plate separation increases,
\[
C \downarrow
\]
Using
\[
Q=CV
\]
\[
Q \downarrow
\]
Thus,
\[
\boxed{
\begin{array}{c}
V \text{ remains constant}
Q \text{ changes}
\end{array}
}
\]
Hence, statement (C) is true and statement (A) is false.
Step 2: Analyse Case-II (key opened).
After opening the key, the capacitor becomes isolated.
Therefore,
\[
Q=\text{constant}
\]
As the plate separation increases,
\[
C \downarrow
\]
Using
\[
V=\frac{Q}{C}
\]
\[
V \uparrow
\]
Thus,
\[
\boxed{
\begin{array}{c}
Q \text{ remains constant}
V \text{ changes}
\end{array}
}
\]
Hence, statement (D) is true and statement (B) is false.
Step 3: Choose the correct statements.
The correct statements are
\[
(C)\ \text{and}\ (D)
\]
\[
\boxed{
(C)\ \text{and}\ (D)\ \text{only}
}
\]
Hence, the correct option is
\[
\boxed{(B)}
\]
