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Physics · Textbook solutions

Electrostatic Potential and Capacitance

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 21 questions

Worked Examples

10 q

Solved Examples

Worked · 10
  1. Eg 2.1
    (a) Calculate the potential at a point P due to a charge of 4×10−7 C4 \times 10^{-7}\,\text{C} located 9 cm9\,\text{cm} away. (b) Hence obtain the work done in bringing a charge of 2×10−9 C2 \times 10^{-9}\,\text{C} from infinity to the point P. Does the answer depend on the path along which the charge is brought?
  2. Eg 2.2
    Two charges 3×10−8 C3 \times 10^{-8}\,\text{C} and −2×10−8 C-2 \times 10^{-8}\,\text{C} are located 15 cm15\,\text{cm} apart. At what point on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.
  3. Eg 2.3
    Figures 2.8 (a) and (b) show the field lines of a positive and negative point charge respectively. [Read from Fig. 2.8: in (a) the field lines radiate outward from the positive charge and the two marked points lie on the same side of it, with P nearer the charge than Q; in (b) the field lines converge inward on the negative charge and the two marked points lie on the same side of it, with A nearer the charge than B.] (a) Give the signs of the potential difference VP−VQV_P - V_Q; VB−VAV_B - V_A. (b) Give the sign of the potential energy difference of a small negative charge between the points Q and P; A and B. (c) Give the sign of the work done by the field in moving a small positive charge from Q to P. (d) Give the sign of the work done by the external agency in moving a small negative charge from B to A. (e) Does the kinetic energy of a small negative charge increase or decrease in going from B to A?
  4. Eg 2.4
    Four charges are arranged at the corners of a square ABCD of side dd, as shown in Fig. 2.15. [Read from Fig. 2.15: A carries +q+q, B carries −q-q, C carries +q+q and D carries −q-q; A and B are the two upper corners, C is below B and D is below A, and E is the centre of the square.] (a) Find the work required to put together this arrangement. (b) A charge q0q_0 is brought to the centre E of the square, the four charges being held fixed at its corners. How much extra work is needed to do this?
  5. Eg 2.5
    (a) Determine the electrostatic potential energy of a system consisting of two charges 7 μC7\,\mu\text{C} and −2 μC-2\,\mu\text{C} (and with no external field) placed at (−9 cm, 0, 0)(-9\,\text{cm},\ 0,\ 0) and (9 cm, 0, 0)(9\,\text{cm},\ 0,\ 0) respectively. (b) How much work is required to separate the two charges infinitely away from each other? (c) Suppose that the same system of charges is now placed in an external electric field E=A (1/r2)E = A\,(1/r^{2}); A=9×105 N C−1 m2A = 9 \times 10^{5}\,\text{N C}^{-1}\,\text{m}^{2}. What would the electrostatic energy of the configuration be?
  6. Eg 2.6
    A molecule of a substance has a permanent electric dipole moment of magnitude 10−29 C m10^{-29}\,\text{C m}. A mole of this substance is polarised (at low temperature) by applying a strong electrostatic field of magnitude 106 V m−110^{6}\,\text{V m}^{-1}. The direction of the field is suddenly changed by an angle of 60∘60^\circ. Estimate the heat released by the substance in aligning its dipoles along the new direction of the field. For simplicity, assume 100% polarisation of the sample.
  7. 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?
  8. Eg 2.8
    A slab of material of dielectric constant KK has the same area as the plates of a parallel-plate capacitor but has a thickness (3/4)d(3/4)d, where dd is the separation of the plates. How is the capacitance changed when the slab is inserted between the plates?
  9. Eg 2.9
    A network of four 10 μF10\,\mu\text{F} capacitors is connected to a 500 V500\,\text{V} supply, as shown in Fig. 2.29. Determine (a) the equivalent capacitance of the network and (b) the charge on each capacitor. (Note, the charge on a capacitor is the charge on the plate with higher potential, equal and opposite to the charge on the plate with lower potential.) [Fig. 2.29 shows the network as follows. The 500 V500\,\text{V} supply is applied across the two terminals A and D. Capacitor C1C_1 is connected between A and B, capacitor C2C_2 between B and C, and capacitor C3C_3 between C and D, so that C1C_1, C2C_2 and C3C_3 form a series chain running from A through B and C to D. Capacitor C4C_4 is connected directly between A and D, in parallel with that series chain.]
  10. Eg 2.10
    (a) A 900 pF900\,\text{pF} capacitor is charged by 100 V100\,\text{V} battery [Fig. 2.31(a)]. How much electrostatic energy is stored by the capacitor? (b) The capacitor is disconnected from the battery and connected to another 900 pF900\,\text{pF} capacitor [Fig. 2.31(b)]. What is the electrostatic energy stored by the system?

Exercises

11 q
  1. Ex 2.1
    Two charges 5×10−8 C5 \times 10^{-8}\,\text{C} and −3×10−8 C-3 \times 10^{-8}\,\text{C} are located 16 cm16\,\text{cm} apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.
  2. Ex 2.2
    A regular hexagon of side 10 cm10\,\text{cm} has a charge 5 μC5\,\mu\text{C} at each of its vertices. Calculate the potential at the centre of the hexagon.
  3. Ex 2.3
    Two charges 2 μC2\,\mu\text{C} and −2 μC-2\,\mu\text{C} are placed at points A and B 6 cm6\,\text{cm} apart. (a) Identify an equipotential surface of the system. (b) What is the direction of the electric field at every point on this surface?
  4. Ex 2.4
    A spherical conductor of radius 12 cm12\,\text{cm} has a charge of 1.6×10−7 C1.6 \times 10^{-7}\,\text{C} distributed uniformly on its surface. What is the electric field (a) inside the sphere (b) just outside the sphere (c) at a point 18 cm18\,\text{cm} from the centre of the sphere?
  5. Ex 2.5
    A parallel plate capacitor with air between the plates has a capacitance of 8 pF8\,\text{pF} (1 pF=10−12 F)(1\,\text{pF} = 10^{-12}\,\text{F}). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 66?
  6. Ex 2.6
    Three capacitors each of capacitance 9 pF9\,\text{pF} are connected in series. (a) What is the total capacitance of the combination? (b) What is the potential difference across each capacitor if the combination is connected to a 120 V120\,\text{V} supply?
  7. Ex 2.7
    Three capacitors of capacitances 2 pF2\,\text{pF}, 3 pF3\,\text{pF} and 4 pF4\,\text{pF} are connected in parallel. (a) What is the total capacitance of the combination? (b) Determine the charge on each capacitor if the combination is connected to a 100 V100\,\text{V} supply.
  8. Ex 2.8
    In a parallel plate capacitor with air between the plates, each plate has an area of 6×10−3 m26 \times 10^{-3}\,\text{m}^2 and the distance between the plates is 3 mm3\,\text{mm}. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V100\,\text{V} supply, what is the charge on each plate of the capacitor?
  9. Ex 2.9
    Explain what would happen if in the capacitor given in Exercise 2.8, a 3 mm3\,\text{mm} thick mica sheet (of dielectric constant =6= 6) were inserted between the plates, (a) while the voltage supply remained connected. (b) after the supply was disconnected.
  10. Ex 2.10
    A 12 pF12\,\text{pF} capacitor is connected to a 50 V50\,\text{V} battery. How much electrostatic energy is stored in the capacitor?
  11. Ex 2.11
    A 600 pF600\,\text{pF} capacitor is charged by a 200 V200\,\text{V} supply. It is then disconnected from the supply and is connected to another uncharged 600 pF600\,\text{pF} capacitor. How much electrostatic energy is lost in the process?