Physics · Textbook solutions

Electrostatic Potential and Capacitance

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 21 questions

Worked Examples

10 q

Solved Examples

Worked · 10

Eg 2.1 $4 \times 10^{-7}\,\text{C}$
Eg 2.2 $3 \times 10^{-8}\,\text{C}$
Eg 2.3 $V_P - V_Q$
Eg 2.4 $d$
Eg 2.5 $7\,\mu\text{C}$
Eg 2.6 $10^{-29}\,\text{C m}$
Eg 2.7 $(a) A comb run through one's dry hair attracts small bits of paper. Why? What happens if the hair is wet or if it is a rainy day? (Remember, a paper does not conduct electricity.) (b) Ordinary rubber is an insulator. But special rubber tyres of aircraft are made slightly conducting. Why is this necessary? (c) Vehicles carrying inflammable materials usually have metallic ropes touching the ground during motion. Why? (d) A bird perches on a bare high power line, and nothing happens to the bird. A man standing on the ground touches the same line and gets a fatal shock. Why?$
Eg 2.8 $K$
Eg 2.9 $10\,\mu\text{F}$
Eg 2.10 $900\,\text{pF}$

Exercises

11 q

Ex 2.1 $5 \times 10^{-8}\,\text{C}$
Ex 2.2 $10\,\text{cm}$
Ex 2.3 $2\,\mu\text{C}$
Ex 2.4 $12\,\text{cm}$
Ex 2.5 $8\,\text{pF}$
Ex 2.6 $9\,\text{pF}$
Ex 2.7 $2\,\text{pF}$
Ex 2.8 $6 \times 10^{-3}\,\text{m}^2$
Ex 2.9 $3\,\text{mm}$
Ex 2.10 $12\,\text{pF}$
Ex 2.11 $600\,\text{pF}$

What the board has asked from this chapter

Questions per paper · 21 papers · 2023 to 2026

Bars average the full papers of each year. 29 further questions come from set variants and are listed below.

2026

26 q

1 $-2q$
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Potential Energy of a System of Charges
1 $V = 10 - 50x$
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Equipotential Surfaces
1 $\vec{E} = 4 x \hat{i}$
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Electrostatic Potential due to Charges and Dipoles
1 $-2q$
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Potential Energy of a System of Charges
2 $r_1$
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Electrostatics of Conductors and Dielectrics
2 $The figure represents the variation of the electric potential V at a point in a region of space as a function of its position along the x-axis. A charged particle will experience the maximum force at :$
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Equipotential Surfaces
2 $r_1$
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Electrostatics of Conductors and Dielectrics
5 $x$
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Equipotential Surfaces
11 $-$
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Electrostatic Potential due to Charges and Dipoles
12 $\mu\mathrm{C}$
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Potential Energy of a System of Charges
22 $4\ \mu\mathrm{F}$
Capacitors, Capacitance and their Combinations
23 $\vec{\mathrm{E}} = \mathrm{E_0}\hat{i}$
Electrostatic Potential due to Charges and Dipoles
23 $A parallel plate capacitor of capacitance C has a dielectric slab between its plates. It is charged to a potential difference V by connecting it across a battery. The battery is then disconnected. If the dielectric slab is now withdrawn from the capacitor, how will the following be affected ? (a) Capacitance of the capacitor, (b) Energy stored in the capacitor, and (c) The potential difference between the plates of the capacitor. Justify your answer in each case.$
Capacitors, Capacitance and their Combinations
25 $\vec{\mathrm{E}}$
Equipotential Surfaces
26 $\mu\mathrm{F}$
Capacitors, Capacitance and their Combinations
26 $(a) Draw electric field lines and equipotential surfaces for a system of two equal positive charges separated by some distance. (b) Using the drawing in (a) above, explain that electric field need not be a constant over each point of an equipotential surface.$
Equipotential Surfaces
26 $A parallel plate capacitor of capacitance C is charged to V volt by a battery. After sometime the battery is disconnected and the distance between the plates is doubled. A slab of dielectric constant k = 1.8 is then introduced to completely fill the space between the plates. How will the following be affected ? (a) The capacitance of the capacitor. (b) The electric field between the plates of the capacitor. (c) The energy stored in the capacitor. Justify your answer in each case.$
Capacitors, Capacitance and their Combinations
The electric potential (V) and electric field (E) are closely related concepts in electrostatics. The electric field is a vector quantity that represents the force per unit charge at a given point in space, whereas electric potential is a scalar quantity that represents the potential energy per unit charge at a given point in space. Electric field and electric potential are related by the equations $\mathrm{E_r} = \dfrac{-dV}{dr}$ and $\vec{\mathrm{E}} = \mathrm{E_r}\hat{r}$ , i.e., electric field is the negative gradient of the electric potential. This means that electric field points in the direction of decreasing potential and its magnitude is the rate of change of potential with distance. The electric field is the force that drives a unit charge to move from higher potential region to lower potential region and electric potential difference between the two points determines the work done in moving a unit charge from one point to the other point. A pair of square conducting plates having sides of length $0{\cdot}05$ m are arranged parallel to each other in x-y plane. They are $0{\cdot}01$ m apart along z-axis and are connected to a 200 V power supply as shown in the figure. An electron enters with a speed of $3 \times 10^7\ \mathrm{ms^{-1}}$ horizontally and symmetrically in the space between the two plates. Neglect the effect of gravity on the electron.
29 (i) $\vec{\mathrm{E}}$
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Equipotential Surfaces
Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance $U = \dfrac{1}{2} CV^2$ , where symbols have their usual meanings. Two capacitors, one of $3\ \mu\mathrm{F}$ and the other of $6\ \mu\mathrm{F}$ , are connected in series in the circuit as shown in the figure, for a long time.
29 (i) $The total capacitance of the circuit is :$
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Capacitors, Capacitance and their Combinations
Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance $U = \dfrac{1}{2} CV^2$ , where symbols have their usual meanings. Two capacitors, one of $3\ \mu\mathrm{F}$ and the other of $6\ \mu\mathrm{F}$ , are connected in series in the circuit as shown in the figure, for a long time.
29 (ii) $10\ \Omega$
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Capacitors, Capacitance and their Combinations
Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance $U = \dfrac{1}{2} CV^2$ , where symbols have their usual meanings. Two capacitors, one of $3\ \mu\mathrm{F}$ and the other of $6\ \mu\mathrm{F}$ , are connected in series in the circuit as shown in the figure, for a long time.
29 (iii) $The potential difference between point A and B is :$
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Capacitors, Capacitance and their Combinations
Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance $U = \dfrac{1}{2} CV^2$ , where symbols have their usual meanings. Two capacitors, one of $3\ \mu\mathrm{F}$ and the other of $6\ \mu\mathrm{F}$ , are connected in series in the circuit as shown in the figure, for a long time.
29 (iv) (a) $6\ \mu\mathrm{F}$
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Capacitors, Capacitance and their Combinations
Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones that are actually needed in a particular application. Two or more capacitors can be grouped in series or in parallel to achieve desired capacitance and voltage. When connected in series, the total capacitance decreases while the voltage rating increases, whereas in parallel connections, the total capacitance increases and maintains the same voltage rating. A capacitor stores energy in the electric field between its plates and stored energy is proportional to the square of the voltage and capacitance $U = \dfrac{1}{2} CV^2$ , where symbols have their usual meanings. Two capacitors, one of $3\ \mu\mathrm{F}$ and the other of $6\ \mu\mathrm{F}$ , are connected in series in the circuit as shown in the figure, for a long time.
29 (iv) (b) $The wire between two capacitors is cut at point P. The current in the circuit will :$
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Capacitors, Capacitance and their Combinations
31 (a) $Q_1$
Electrostatic Potential due to Charges and Dipoles
32 (a) $\mathrm{C_1}$
Capacitors, Capacitance and their Combinations
33 (a) $-4q$
Potential Energy of a System of Charges

2025

39 q

1 $\mu\mathrm{F}$
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Capacitors, Capacitance and their Combinations
1 $The electric field (E) and electric potential (V) at a point inside a charged hollow metallic sphere are respectively :$
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Electrostatics of Conductors and Dielectrics
1 $A metal sheet is inserted between the plates of a parallel plate capacitor of capacitance C. If the sheet partly occupies the space between the plates, the capacitance :$
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Electrostatics of Conductors and Dielectrics
1 $l$
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Electrostatics of Conductors and Dielectrics
1 $In the figure curved lines represent equipotential surfaces. A charge Q is moved along different paths A, B, C and D. The work done on the charge will be maximum along the path$
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Equipotential Surfaces
1 $Two horizontal plates, separated by 1 cm, are arranged one above the other. A particle of mass 5 mg and charge 2 nC is released in air between the plates. The potential difference that should be applied to the plates so that the particle remains suspended between them, is :$
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Equipotential Surfaces
2 $The effective capacitance of the network between points A and B shown in the figure is :$
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Capacitors, Capacitance and their Combinations
2 $\vec{E} = \alpha \dfrac{\vec{r}}{|\vec{r}|^3}$
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Electrostatic Potential due to Charges and Dipoles
3 $l$
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Potential Energy of a System of Charges
22 $(a) "There is a limit to the amount of charge that can be stored on a given capacitor." Explain. (b) A capacitor is charged by a battery to a potential difference V. It is disconnected from the battery and connected across another identical uncharged capacitor. Calculate the ratio of total energy stored in the combination to the initial energy stored in the capacitor.$
Energy Stored in a Capacitor
23 $A parallel plate capacitor has plate area A and plate separation d. Half of the space between the plates is filled with a material of dielectric constant K in two ways as shown in the figure. Find the values of the capacitance of the capacitors in the two cases.$
Capacitors, Capacitance and their Combinations
24 $A capacitor of plate area A and plate separation d is charged by a battery to voltage V. The battery is disconnected and plates are slowly pulled apart till the separation becomes 2d. Find the value of : (a) potential difference between the plates, (b) electric field between the plates, (c) work done in pulling the plates apart.$
Capacitors, Capacitance and their Combinations
28 (a) $2\sigma$
Electrostatics of Conductors and Dielectrics
28 $-5\ \mu\mathrm{C}$
Potential Energy of a System of Charges
A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.
29 (i) $The electric field between the plates of a parallel plate capacitor is E. Now the separation between the plates is doubled and simultaneously the applied potential difference between the plates is reduced to half of its initial value. The new value of the electric field between the plates will be :$
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Capacitors, Capacitance and their Combinations
A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key $\mathrm{S_1}$ (keeping $\mathrm{S_2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c$ (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as $q = Q[1 - e^{-t/RC}]$ The charging current can be obtained by differentiating it and using $\frac{d}{dx}(e^{mx}) = m e^{mx}$ . Consider the case when R = 20 k $\Omega$ , C = 500 $\mu$ F and V = 10 V.
29 (i) $\mathrm{S_1}$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side 'L' held parallel and equidistant from each other. The space between $P_1$ and $P_2$ and $P_2$ and $P_3$ is completely filled with mica sheets of dielectric constant 'K'. The plate $P_2$ is connected to point A and other plates $P_1$ and $P_3$ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.
29 (i) $The capacitance of the system between A and B will be :$
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Capacitors, Capacitance and their Combinations
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 $V = V_1 - V_2$ between them. The ratio $\dfrac{Q}{V}$ 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 $E_0$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\sigma$ ), 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 $\sigma_p$ . For a linear dielectric $\sigma_p$ is proportional to $E_0$ . 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.
29 (i) $Consider a capacitor of capacitance C, with plate area A and plate separation d, filled with air [Fig. (a)]. The distance between the plates is increased to 2d and one of the plates is shifted as shown in Fig. (b). The capacitance of the new system now is :$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side 'L' held parallel and equidistant from each other. The space between $P_1$ and $P_2$ and $P_2$ and $P_3$ is completely filled with mica sheets of dielectric constant 'K'. The plate $P_2$ is connected to point A and other plates $P_1$ and $P_3$ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.
29 (ii) $P_1$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.
29 (ii) $V$
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Capacitors, Capacitance and their Combinations
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 $V = V_1 - V_2$ between them. The ratio $\dfrac{Q}{V}$ 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 $E_0$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\sigma$ ), 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 $\sigma_p$ . For a linear dielectric $\sigma_p$ is proportional to $E_0$ . 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.
29 (ii) $d_1$
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Electrostatics of Conductors and Dielectrics
A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key $\mathrm{S_1}$ (keeping $\mathrm{S_2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c$ (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as $q = Q[1 - e^{-t/RC}]$ The charging current can be obtained by differentiating it and using $\frac{d}{dx}(e^{mx}) = m e^{mx}$ . Consider the case when R = 20 k $\Omega$ , C = 500 $\mu$ F and V = 10 V.
29 (ii) $\mathrm{S_1}$
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Capacitors, Capacitance and their Combinations
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 $V = V_1 - V_2$ between them. The ratio $\dfrac{Q}{V}$ 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 $E_0$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\sigma$ ), 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 $\sigma_p$ . For a linear dielectric $\sigma_p$ is proportional to $E_0$ . 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.
29 (iii) $An electric field E is established between the plates of an air filled parallel plate capacitor, with charges Q and - Q. V is the volume of the space enclosed between the plates. The energy stored in the capacitor is :$
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Energy Stored in a Capacitor
A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side 'L' held parallel and equidistant from each other. The space between $P_1$ and $P_2$ and $P_2$ and $P_3$ is completely filled with mica sheets of dielectric constant 'K'. The plate $P_2$ is connected to point A and other plates $P_1$ and $P_3$ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.
29 (iii) $P_1$
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Capacitors, Capacitance and their Combinations
A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key $\mathrm{S_1}$ (keeping $\mathrm{S_2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c$ (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as $q = Q[1 - e^{-t/RC}]$ The charging current can be obtained by differentiating it and using $\frac{d}{dx}(e^{mx}) = m e^{mx}$ . Consider the case when R = 20 k $\Omega$ , C = 500 $\mu$ F and V = 10 V.
29 (iii) $The dimensional formula for RC is$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.
29 (iii) $In the above figure P, Q are the two parallel plates of a capacitor. Plate Q is at positive potential with respect to plate P. MN is an imaginary line drawn perpendicular to the plates. Which of the graphs shows correctly the variations of the magnitude of electric field strength E along the line MN ?$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.
29 (iv) (a) $\vec{d}$
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Capacitors, Capacitance and their Combinations
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 $V = V_1 - V_2$ between them. The ratio $\dfrac{Q}{V}$ 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 $E_0$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\sigma$ ), 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 $\sigma_p$ . For a linear dielectric $\sigma_p$ is proportional to $E_0$ . 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.
29 (iv) (a) $Q'$
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Capacitors, Capacitance and their Combinations
A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key $\mathrm{S_1}$ (keeping $\mathrm{S_2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c$ (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as $q = Q[1 - e^{-t/RC}]$ The charging current can be obtained by differentiating it and using $\frac{d}{dx}(e^{mx}) = m e^{mx}$ . Consider the case when R = 20 k $\Omega$ , C = 500 $\mu$ F and V = 10 V.
29 (iv) (a) $\mathrm{S_1}$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side 'L' held parallel and equidistant from each other. The space between $P_1$ and $P_2$ and $P_2$ and $P_3$ is completely filled with mica sheets of dielectric constant 'K'. The plate $P_2$ is connected to point A and other plates $P_1$ and $P_3$ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.
29 (iv) (a) $L^2$
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Capacitors, Capacitance and their Combinations
A circuit consisting of a capacitor C, a resistor of resistance R and an ideal battery of emf V, as shown in figure is known as RC series circuit. As soon as the circuit is completed by closing key $\mathrm{S_1}$ (keeping $\mathrm{S_2}$ open) charges begin to flow between the capacitor plates and the battery terminals. The charge on the capacitor increases and consequently the potential difference $V_c$ (= q/C) across the capacitor also increases with time. When this potential difference equals the potential difference across the battery, the capacitor is fully charged (Q = VC). During this process of charging, the charge q on the capacitor changes with time t as $q = Q[1 - e^{-t/RC}]$ The charging current can be obtained by differentiating it and using $\frac{d}{dx}(e^{mx}) = m e^{mx}$ . Consider the case when R = 20 k $\Omega$ , C = 500 $\mu$ F and V = 10 V.
29 (iv) (b) $\mathrm{S_1}$
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Capacitors, Capacitance and their Combinations
A parallel plate capacitor has two parallel plates which are separated by an insulating medium like air, mica, etc. When the plates are connected to the terminals of a battery, they get equal and opposite charges and an electric field is set up in between them. This electric field between the two plates depends upon the potential difference applied, the separation of the plates and nature of the medium between the plates.
29 (iv) (b) $A material of dielectric constant K is filled in a parallel plate capacitor of capacitance C. The new value of its capacitance becomes$
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Electrostatics of Conductors and Dielectrics
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 $V = V_1 - V_2$ between them. The ratio $\dfrac{Q}{V}$ 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 $E_0$ is localised between the plates and is uniform throughout. When a slab of a dielectric is inserted between the charged plates (charge density $\sigma$ ), 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 $\sigma_p$ . For a linear dielectric $\sigma_p$ is proportional to $E_0$ . 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.
29 (iv) (b) $\dfrac{d}{2}$
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Electrostatics of Conductors and Dielectrics
A parallel plate capacitor consists of two conducting plates kept generally parallel to each other at a distance. When the capacitor is charged, the charge resides on the inner surfaces of the plates and an electric field is set up between them. Thus, electrostatic energy is stored in the capacitor. The figure shows three large square metallic plates, each of side 'L' held parallel and equidistant from each other. The space between $P_1$ and $P_2$ and $P_2$ and $P_3$ is completely filled with mica sheets of dielectric constant 'K'. The plate $P_2$ is connected to point A and other plates $P_1$ and $P_3$ are connected to point B. Point A is maintained at a positive potential with respect to point B and the potential difference between A and B is V.
29 (iv) (b) $If the source of potential difference applied between A and B is removed, and then A and B are connected by a conducting wire, the net charge on the system will be :$
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Capacitors, Capacitance and their Combinations
31 (a) $\mu$
Potential Energy of a System of Charges
31 (a) $^{-1}$
Electrostatic Potential due to Charges and Dipoles
31 (a) $\mathrm{C_0}$
Capacitors, Capacitance and their Combinations
31 (b) $\vec{p} = (0{\cdot}8\,\hat{i} + 0{\cdot}6\,\hat{j})10^{-29}$
Electrostatic Potential due to Charges and Dipoles
33 (a) $\vec{E} = 40x\,\hat{i}$
Electrostatic Potential due to Charges and Dipoles

2024

29 q

1 $P_1$
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Capacitors, Capacitance and their Combinations
1 $q_1$
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Equipotential Surfaces
1 $+q$
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Potential Energy of a System of Charges
1 $Three point charges, each of charge q are placed on vertices of a triangle ABC, with AB = AC = 5L, BC = 6L. The electrostatic potential at midpoint of side BC will be$
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Electrostatic Potential due to Charges and Dipoles
1 $An isolated conductor, with a cavity, has a net charge +Q. A point charge +q is inside the cavity. The charges on the cavity wall and the outer surface are respectively :$
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Electrostatics of Conductors and Dielectrics
1 $The capacitance of a parallel plate capacitor having a medium of dielectric constant K = 4 in between the plates is C. If this medium is removed, then the capacitance of the capacitor becomes :$
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Capacitors, Capacitance and their Combinations
2 $P_1$
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Electrostatic Potential due to Charges and Dipoles
2 $\mu\mathrm{F}$
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Energy Stored in a Capacitor
22 $\mathrm{Q_1}$
Potential Energy of a System of Charges
22 $4{\cdot}0\ \mu\mathrm{C}$
Energy Stored in a Capacitor
23 $\vec{E} = (10x + 4)\hat{i}$
Electrostatic Potential due to Charges and Dipoles
23 $A thin spherical conducting shell of radius R has a charge q. A point charge Q is placed at the centre of the shell. Find (i) The charge density on the outer surface of the shell and (ii) the potential at a distance of (R/2) from the centre of the shell.$
Electrostatics of Conductors and Dielectrics
23 $\sigma$
Electrostatics of Conductors and Dielectrics
Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric. The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.
29 (i) $Which of the following is a polar molecule ?$
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Electrostatics of Conductors and Dielectrics
The figure shows four pairs of parallel identical conducting plates, separated by the same distance $2{\cdot}0$ cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.
29 (i) $\vec{E}$
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Equipotential Surfaces
Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric. The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.
29 (ii) $Which of the following statements about dielectrics is correct ?$
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Electrostatics of Conductors and Dielectrics
The figure shows four pairs of parallel identical conducting plates, separated by the same distance $2{\cdot}0$ cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.
29 (ii) $An electron is released midway between the plates of pair IV. It will :$
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Equipotential Surfaces
The figure shows four pairs of parallel identical conducting plates, separated by the same distance $2{\cdot}0$ cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.
29 (iii) $V_0$
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Equipotential Surfaces
Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric. The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.
29 (iii) $When a dielectric slab is inserted between the plates of an isolated charged capacitor, the energy stored in it :$
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Energy Stored in a Capacitor
Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric. The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.
29 (iv) (a) $C_0$
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Capacitors, Capacitance and their Combinations
The figure shows four pairs of parallel identical conducting plates, separated by the same distance $2{\cdot}0$ cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.
29 (iv) (a) $E_1$
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Equipotential Surfaces
The figure shows four pairs of parallel identical conducting plates, separated by the same distance $2{\cdot}0$ cm and arranged perpendicular to x-axis. The electric potential of each plate is mentioned. The electric field between a pair of plates is uniform and normal to the plates.
29 (iv) (b) $1{\cdot}76 \times 10^{11}$
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Electrostatic Potential due to Charges and Dipoles
Dielectrics play an important role in design of capacitors. The molecules of a dielectric may be polar or non-polar. When a dielectric slab is placed in an external electric field, opposite charges appear on the two surfaces of the slab perpendicular to electric field. Due to this an electric field is established inside the dielectric. The capacitance of a capacitor is determined by the dielectric constant of the material that fills the space between the plates. Consequently, the energy storage capacity of a capacitor is also affected. Like resistors, capacitors can also be arranged in series and/or parallel.
29 (iv) (b) $2\ C_0$
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Capacitors, Capacitance and their Combinations
31 (a) $\mu$
Capacitors, Capacitance and their Combinations
31 (a) $\vec{p}$
Electrostatic Potential due to Charges and Dipoles
31 (a) $\vec{p}$
Potential Energy of a System of Charges
31 (a) $q_1$
Potential Energy of a System of Charges
31 (b) $\vec{p} = p\,\hat{i}$
Electrostatic Potential due to Charges and Dipoles
32 (a) $(i) A dielectric slab of dielectric constant 'K' and thickness 't' is inserted between plates of a parallel plate capacitor of plate separation d and plate area A. Obtain an expression for its capacitance. (ii) Two capacitors of different capacitances are connected first (1) in series and then (2) in parallel across a dc source of 100 V. If the total energy stored in the combination in the two cases are 40 mJ and 250 mJ respectively, find the capacitance of the capacitors.$
Capacitors, Capacitance and their Combinations

2023

18 q

2 $x$
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Electrostatic Potential due to Charges and Dipoles
12 $\mu\mathrm{F}$
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Capacitors, Capacitance and their Combinations
In question number 16 to 18, two statements are given - one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer to these questions from the codes (a), (b), (c) and (d) as given below :
17 $Assertion (A) : Work done in moving a charge around a closed path, in an electric field is always zero. Reason (R) : Electrostatic force is a conservative force.$
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Electrostatic Potential due to Charges and Dipoles
22 $-q$
Potential Energy of a System of Charges
25 (a) $q$
Potential Energy of a System of Charges
25 (b) $r_1$
Electrostatics of Conductors and Dielectrics
26 (a) $Two charged conducting spheres of radii a and b are connected to each other by a wire. Find the ratio of the electric fields at their surfaces.$
Electrostatics of Conductors and Dielectrics
26 $(a) Twelve negative charges of same magnitude are equally spaced and fixed on the circumference of a circle of radius R as shown in Fig. (i). Relative to potential being zero at infinity, find the electric potential and electric field at the centre C of the circle. (b) If the charges are unequally spaced and fixed on an arc of 120° of radius R as shown in Fig. (ii), find electric potential at the centre C.$
Electrostatic Potential due to Charges and Dipoles
26 (b) $A parallel plate capacitor (A) of capacitance C is charged by a battery to voltage V. The battery is disconnected and an uncharged capacitor (B) of capacitance 2C is connected across A. Find the ratio of (i) final charges on A and B. (ii) total electrostatic energy stored in A and B finally and that stored in A initially.$
Energy Stored in a Capacitor
28 $\mathrm{Q_1}$
Potential Energy of a System of Charges
30 $\mu\mathrm{F}$
Energy Stored in a Capacitor
31 (b) $(i) Consider two identical point charges located at points (0, 0) and (a, 0). (1) Is there a point on the line joining them at which the electric field is zero ? (2) Is there a point on the line joining them at which the electric potential is zero ? Justify your answers for each case. (ii) State the significance of negative value of electrostatic potential energy of a system of charges. Three charges are placed at the corners of an equilateral triangle ABC of side 2.0 m as shown in figure. Calculate the electric potential energy of the system of three charges.$
Potential Energy of a System of Charges
31 (b) $(i) Define electric potential at a point and write its SI unit. (ii) Two capacitors are connected in series. Derive an expression of the equivalent capacitance of the combination. (iii) Two point charges + q and - q are located at points (3a, 0) and (0, 4a) respectively in x-y plane. A third charge Q is kept at the origin. Find the value of Q, in terms of q and a, so that the electrostatic potential energy of the system is zero.$
Capacitors, Capacitance and their Combinations
Electrostatics deals with the study of forces, fields and potentials arising from static charges. Force and electric field, due to a point charge is basically determined by Coulomb's law. For symmetric charge configurations, Gauss's law, which is also based on Coulomb's law, helps us to find the electric field. A charge/a system of charges like a dipole experience a force/torque in an electric field. Work is required to be done to provide a specific orientation to a dipole with respect to an electric field.
34 (b) $\dfrac{1}{r}$
Electrostatic Potential due to Charges and Dipoles
A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.
35 (i) $Find the equivalent capacitance between points A and B in the given diagram.$
Capacitors, Capacitance and their Combinations
A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.
35 (ii) $A dielectric slab is inserted between the plates of a parallel plate capacitor. The electric field between the plates decreases. Explain.$
Electrostatics of Conductors and Dielectrics
A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.
35 (iii) (a) $A capacitor A of capacitance C, having charge Q is connected across another uncharged capacitor B of capacitance 2C. Find an expression for (a) the potential difference across the combination and (b) the charge lost by capacitor A.$
Capacitors, Capacitance and their Combinations
A capacitor is a system of two conductors separated by an insulator. The two conductors have equal and opposite charges with a potential difference between them. The capacitance of a capacitor depends on the geometrical configuration (shape, size and separation) of the system and also on the nature of the insulator separating the two conductors. They are used to store charges. Like resistors, capacitors can be arranged in series or parallel or a combination of both to obtain desired value of capacitance.
35 (iii) (b) $Two slabs of dielectric constants 2K and K fill the space between the plates of a parallel plate capacitor of plate area A and plate separation d as shown in figure. Find an expression for capacitance of the system.$
Electrostatics of Conductors and Dielectrics

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 2 times: 2023, 2026
2 $r_1$
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Question 2
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
1 $-2q$
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Question 3
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
1 $-2q$
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Question 4
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
23 $\vec{\mathrm{E}} = \mathrm{E_0}\hat{i}$
Question 5
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
2 $r_1$
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Question 6
This type asked 3 times: 2023, 2025, 2026. This one: 2026
26 $\mu\mathrm{F}$
Question 7
This type asked 3 times: 2024, 2025, 2026. This one: 2026
1 $\vec{E} = 4 x \hat{i}$
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Question 8
This type asked 3 times: 2023, 2024, 2026. This one: 2026
12 $\mu\mathrm{C}$
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Question 9
This type asked 2 times: 2025, 2026. This one: 2026
31 (a) $Q_1$
Question 10
This type asked 2 times: 2024, 2026. This one: 2026
26 $(a) Draw electric field lines and equipotential surfaces for a system of two equal positive charges separated by some distance. (b) Using the drawing in (a) above, explain that electric field need not be a constant over each point of an equipotential surface.$

Worked Examples

Solved Examples

Exercises

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10