Q33 of 38 Page 88

A parallel plate capacitor is to be designed with a voltage rating 1 kV, using a material of 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 never to exceed, say 10% of the dielectric strength.

What minimum area of the plates is required to have a capacitance of 50 pF?

Potential difference of a parallel plate capacitor(V) = 1kV = 1000V


Dielectric constant of a material(ϵr) = 3


Dielectric strength = 107V/m


Electric field intensity(E) = 10%of 107


⇒ E = 106V/m


(since, the field intensity never exceeds 10% of the dielectric strength)


Capacitance of the parallel plate capacitor(C) = 50pF = 50 × 10-12F


Distance between the plates (d),


⇒


⇒ d = 10-3m


∵ Capacitance =


where, ϵ0 = Permittivity of free space = 8.85 × 10-12N-1 m-2 C2


A = area of plates


∴


⇒ A = 50 × 10-12F × 10-3m/8.85 × 10-12N-1 m-2 C2 × 3


⇒ A = 1.88 × 10-3m2


∴ Minimum area of the plates required = 1.88 × 10-3m2


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31

Answer carefully:

(a) Two large conducting spheres carrying charges Q1 and Q2 are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q1 Q2/4πε0r2, where r is the distance between their centres?


(b) If Coulomb’s law involved 1/r3 dependence (instead of 1/r2), would Gauss’s law be still true?


(c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point?


(d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical?


(e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there?


(f) What meaning would you give to the capacitance of a single conductor?


(g) Guess a possible reason why water has a much greater dielectric constant ( = 80) than say, mica ( = 6).

 32

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. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).

 34

Describe schematically the equipotential surfaces corresponding to

(a) a constant electric field in the z-direction,


(b) a field that uniformly increases in magnitude but remains in a constant (say, z) direction,


(c) a single positive charge at the origin, and


(d) a uniform grid consisting of long equally spaced parallel charged wires in a plane.

 35

In a Van de Graff 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.)