Two charges 5 × 10^-8 C and -3 × 10^-8 C are located 16 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.

A regular hexagon of side 10 cm has a charge 5 µC at each of its vertices. Calculate the potential at the centre of the hexagon.

Two charges 2 µC and -2 µC are placed at points A and B 6 cm apart. (I) Identify an equipotential surface of thesystem. (II) What is the direction of the electric field at every point on thissurface?

A spherical conductor of radius 12 cm has a charge of 1.6 × 10^-7C distributed uniformly on its surface. What is the electric field (I) Inside thesphere (II) Just outside thesphere (III) At a point 18 cm from the centre of thesphere?

Aparallelplatecapacitorwithairbetweentheplates hasacapacitanceof8pF(1pF = 10^-12F).Whatwillbethecapacitanceifthedistancebetweentheplatesisreducedby half,andthespacebetweenthemisfilledwithasubstanceofdielectricconstant6?

Three capacitors each of capacitance 9 pF are connected in series. (I) What is the total capacitance of thecombination? (II) Whatisthepotentialdifferenceacrosseachcapacitorifthecombinationis connected to a 120 Vsupply?

Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel. (I) What is the total capacitance of thecombination? (II) Determinethechargeoneachcapacitorifthecombinationisconnectedtoa100V supply.

In a parallel plate capacitor with air between the plates, each plate has an area of 6 × 10^-3 m² and the distance between the plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V supply, what is the charge on each plate of the capacitor?

Explain what would happen if in the capacitor given in Exercise 2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates, (I) While the voltage supply remainedconnected. (II) After the supply wasdisconnected.

A12pFcapacitorisconnectedtoa50Vbattery.Howmuchelectrostaticenergyisstored in thecapacitor?

A600pFcapacitorischargedbya200 Vsupply.Itisthendisconnectedfromthesupply andisconnectedtoanotheruncharged600pF capacitor.Howmuchelectrostaticenergy is lost in theprocess?

A charge of 8 mC is located at the origin. Calculate the work done in taking a small charge of -2 × 10^-9 C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R(0, 6 cm, 9 cm).

A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.

Two tiny spheres carrying charges 1.5 µC and 2.5 µC are located 30 cm apart. Find the potential and electric field: (I) at the mid-point of the line joining the two charges,and (II) atapoint10cmfromthismidpoint inaplanenormaltothelineandpassing through themid-point.

A spherical conducting shell of inner radius r1 and outer radius r2 has a charge Q. (I) Achargeqisplacedatthecentreoftheshell.Whatisthesurfacechargedensityon the inner and outer surfaces of theshell? (II) Istheelectricfieldinsideacavity(withno charge)zero,eveniftheshellisnot spherical, but has any irregular shape?Explain.

(I) Showthatthenormalcomponentofelectrostaticfieldhasadiscontinuityfromone side of a charged surface to another givenby Whereis a unit vector normal to the surface at a point and s is the surface charge density at that point. (Thedirectionofis from side 1 to side 2.) Hence show thatjust outside a conductor, the electric field is s (II) Showthatthetangentialcomponentofelectrostaticfieldiscontinuousfromoneside of a charged surface to another. [Hint: For (a), use Gauss’s law. For, (b) use the fact that work done by electrostatic field on a closed loop iszero.]

A long charged cylinder of linear charged density ? is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å: (I) EstimatethepotentialenergyofthesystemineV,takingthezeroofthepotential energy at infinite separation of the electron fromproton. (II) Whatistheminimumworkrequiredtofreetheelectron,giventhatitskinetic energy in the orbit is half the magnitude of potential energy obtained in(a)? (III) Whataretheanswers to(a)and(b)aboveifthezeroofpotentialenergyistakenat1.06 Å separation?

Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.

Two charges -q and +q are located at points (0, 0, - a) and (0, 0, a), respectively. (I) What is the electrostatic potential at thepoints? (II) Obtainthedependenceofpotentialonthedistancerofapointfromtheoriginwhen r/a >>1. (III) Howmuchwork isdoneinmovingasmalltestchargefromthepoint(5,0,0)to (-7, 0, 0) along the x-axis? Does the answer change if the path of the test charge between the same points is not along thex-axis?

Figure 2.34 shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

An electrical technician requires a capacitance of 2 µF in a circuit across a potential difference of 1 kV. A large number of 1 µF capacitors are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors.

What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realize from your answer why ordinary capacitors are in the range of µF or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.]

Obtain the equivalent capacitance of the network in Fig. 2.35. For a 300 V supply, determine the charge and voltage across each capacitor.

The plates of a parallel plate capacitor have an area of 90 cm² each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply. (I) How much electrostatic energy is stored by thecapacitor? (II) View this energy as stored in the electrostatic field between the plates, and obtain theenergyperunitvolumeu.Hencearriveatarelationbetweenuandthemagnitudeof electric field E between theplates.

A 4 µF capacitor is charged by a 200 V supply. It is then disconnected from the supply, and is connected to another uncharged 2 µF capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?

Showthattheforceoneachplateofaparallelplatecapacitorhasamagnitudeequalto (½) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor½.

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show that the capacitance of a spherical capacitor is given by where r1 and r2 are the radii of outer and inner spheres, respectively.

A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µC. The space between the concentric spheres is filled with a liquid of dielectric constant 32. (I) Determine the capacitance of thecapacitor. (II) What is the potential of the innersphere? (III) Comparethecapacitanceofthiscapacitorwiththatofanisolatedsphereofradius 12 cm. Explain why the latter is muchsmaller.

A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 µC. Determinethecapacitanceofthesystemandthepotentialoftheinnercylinder.Neglect end effects (i.e., bending of field lines at theends).

Aparallelplatecapacitoristobedesignedwithavoltagerating1kV,usingamaterialof dielectric constant 3 and dielectric strength about 10⁷ Vm^-1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field nevertoexceed,say10%ofthedielectricstrength.Whatminimumareaoftheplatesis required to have a capacitance of 50pF?

Describe schematically the equipotential surfaces corresponding to (I) a constant electric field in thez-direction, (II) afieldthatuniformlyincreasesinmagnitudebutremainsinaconstant(say,z) direction, (III) a single positive charge at the origin,and (IV) a uniform grid consisting of long equally spaced parallel charged wires in aplane.

In a Van de Graaff type generator a spherical metal shell is to be a 15 × 10⁶ V electrode. The dielectric strength of the gas surrounding the electrode is 5 × 10⁷ Vm^-1. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.)

A small sphere of radius r₁ and charge q₁ is enclosed by a spherical shell of radius r₂ and charge q₂. Show that if q₁ is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q₂ on the shell is.