CBSE 2025 · Region 6 · Set 1 · Q29 · 4 marks

A capacitor is a system of two conductors separated by an insulator. In practice, the two conductors have charges Q and -Q with potential difference $\displaystyle \mathrm{V}=\mathrm{V}_{1}-\mathrm{V}_{2}$ between them. The ratio $\displaystyle \frac{\mathrm{Q}}{\mathrm{V}}$ is a constant, denoted by $\displaystyle \mathrm{C}$ and is called the capacitance of the capacitor. It is independent of Q or V. It depends only on the geometrical configuration (shape, size, separation) of the two conductors and the medium separating the conductors. When a parallel plate capacitor is charged, the electric field $\displaystyle \mathrm{E}_{0}$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\displaystyle \sigma$ ) , the dielectric is polarised by the field. Consequently opposite charges appear on the faces of the slab, near the plates, with surface charge density of magnitude $\displaystyle \sigma_{\mathrm{p}}$. For a linear dielectric $\displaystyle \sigma_{\mathrm{p}}$ is proportional to $\displaystyle \mathrm{E}_{0}$. Introduction of a dielectric changes the electric field, and hence, the capacitance of a capacitor, and hence, the energy stored in the capacitor. Like resistors, capacitors can also be arranged in series or in parallel or in a combination of series and parallel.

(i)

Consider a capacitor of capacitance C, with plate area A and plate separation d, filled with air[Fig. (a) ]. The distance between the plates is increased to $\displaystyle 2$ d and one of the plates is shifted as shown in Fig. (b) . The capacitance of the new system now is:

(i)

(a)

(A)

$\displaystyle \frac{\mathrm{C}}{4}$

(B)

$\displaystyle \frac{\mathrm{C}}{2}$

(C)

$\displaystyle 2$ C

(D)

$\displaystyle 4$ C

(ii)

A slab (area A and thickness $\displaystyle \mathrm{d}_{1}$ ) of a linear dielectric of dielectric constant K is inserted between charged plates (charge density $\displaystyle \sigma$ ) of a parallel plate capacitor [plate area A and plate separation $\displaystyle \left.\mathrm{d}\left(>\mathrm{d}_{1}\right)\right]$ and opposite charges with charge density of magnitude $\displaystyle \sigma_{\mathrm{p}}$ appear on the faces of the slab. The dielectric constant K is given by :

(A)

$\displaystyle \frac{\sigma+\sigma_{\mathrm{p}}}{\sigma}$

(B)

$\displaystyle \frac{\sigma}{\sigma-\sigma_{\mathrm{p}}}$

(C)

$\displaystyle \frac{\sigma+\sigma_{\mathrm{p}}}{\sigma_{\mathrm{p}}}$

(D)

$\displaystyle \frac{\sigma}{\sigma_{\mathrm{p}}}$

(iii)

An electric field E is established between the plates of an air filled parallel plate capacitor, with charges Q and $\displaystyle -\mathrm{Q} . \mathrm{V}$ is the volume of the space enclosed between the plates. The energy stored in the capacitor is :

(A)

$\displaystyle \frac{1}{2} \varepsilon_{0} \mathrm{E}^{2}$

(B)

$\displaystyle \varepsilon_{0} \mathrm{Q}^{2} \mathrm{E}$

(C)

$\displaystyle \frac{1}{2} \varepsilon_{0} \mathrm{E}^{2} \mathrm{~V}$

(D)

$\displaystyle \varepsilon_{0} \mathrm{EQV}$

(a)

Three capacitors A, B and M, each of capacitance C are connected to a capacitor N of capacitance $\displaystyle 2$ C and a battery as shown in the figure. If the charges on A and N are Q and $\displaystyle \mathrm{Q}^{\prime}$ respectively, then $\displaystyle \frac{\mathrm{Q}^{\prime}}{\mathrm{Q}}$ is :

(a)

(A)

$\displaystyle \frac{1}{6}$

(B)

$\displaystyle \frac{1}{3}$

(C)

$\displaystyle 3$ (D) $\displaystyle 6$

OR

(b) A slab (area A and thickness $\displaystyle \frac{\mathrm{d}}{2}$ ) of dielectric constant K is inserted in a parallel plate capacitor of plate area A and plate separation d . If C and $\displaystyle \mathrm{C}_{0}$ are the capacitances of the capacitors with and without the dielectric, then $\displaystyle \frac{\mathrm{C}}{\mathrm{C}_{0}}$ is :

(A)

$\displaystyle \frac{\mathrm{K}+1}{2 \mathrm{~K}}$

(B)

$\displaystyle \frac{2 \mathrm{~K}}{\mathrm{~K}+1}$

(C)

$\displaystyle \frac{\mathrm{K}}{\mathrm{K}-1}$

(D)

$\displaystyle \frac{\mathrm{K}-1}{\mathrm{~K}}$

CBSE Class 12 Physics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.