Q30 of 38 Page 88

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.

(a) Determine the capacitance of the capacitor.


(b) What is the potential of the inner sphere?


(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.

Radius of the outer shell (r1) = 13cm = 0.13m


Radius of the inner shell (r2) = 12cm = 0.12m


Charge on the outer surface of the inner shell = 2.5 μC = 2.510-6C


Dielectric constant of liquid () = 32


Since, Potential difference between the two shells,



∴ Δ


Where, = Absolute Permittivity of free space = 8.8510-12C2N-1m-2


Therefore




(a) Capacitance,



⇒ C = 5.5510-9F


Hence, the capacitance of the capacitor = 5.5510-9F


(b) The potential of a capacitor (V), is given as:



⇒


⇒ V = 450 V


Hence, the potential of the inner sphere = 450V


(c) Radius of the isolated sphere(R) = 12 cm = 0.12m


Capacitance of an isolated sphere = 4πε0R


∴ C = 4 × 3.14 × 8.85 × 10-12 × 0.12


C = 1.3310-11F


Where, ϵ0 is the Absolute Permittivity of free space = 8.8510-12C2N-1m-2The capacitance of the isolated sphere is less than that of the concentric spheres because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.


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28

Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (1/2) 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 factor 1/2.

 29

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.


 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).