$7$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 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) (a) Three capacitors A , B and M , each of capacitance C are connected to a capacitor N of capacitance 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 : 2025_CBSE_Physics_Set_Region6_Subset_Set3_QUNKNOWN__i_p21_8

2025_CBSE_Physics_Set_Region6_Subset_Set3_QUNKNOWN__i_p21_8

(A) $\displaystyle \frac{1}{6}$ (B) $\displaystyle \frac{1}{3}$ (C) 3 (D) 6

OR

(b) A slab (area A and thickness $\displaystyle \frac{\mathrm{d}}{2}$ ) of dielectric constant $\displaystyle \mathrm{K}$ 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) $\displaystyle \frac{\mathrm{K}-1}{\mathrm{~K}}$ (ii) 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) $\displaystyle \varepsilon_{0} \mathrm{EQV}$ (iii) 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) $\displaystyle \frac{\sigma}{\sigma_{\mathrm{p}}}$ (iv) 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 2d and one of the plates is shifted as shown in Fig. (b) . The capacitance of the new system now is: 2025_CBSE_Physics_Set_Region6_Subset_Set3_QUNKNOWN__iv_p23_27

2025_CBSE_Physics_Set_Region6_Subset_Set3_QUNKNOWN__iv_p23_27

(a)

2025_CBSE_Physics_Set_Region6_Subset_Set3_QUNKNOWN__iv_p23_30

(b) (A) $\displaystyle \frac{\mathrm{C}}{4}$ (B) $\displaystyle \frac{\mathrm{C}}{2}$ (C) 2 C (D) 4 C

Physics · Electrostatic Potential and Capacitance · Combination of Capacitors