question_answer 29)

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

Answer:

It consists of two concentric spherical shells A and B of radii a and b with charge +q and charge ?q respectively.                 (Outer sphere is grounded).                                                 Capacity of the Spherical Capacitor.                 Draw a Gaussian surface (shown dotted) of radius r (figure)                 Step 1: Finding the Electric field E                 According of Gauss? theorem                                   or                                            are along same line so                 Since E is uniform over the Gaussian surface, so                                 Now  = surface area of Gaussian surface = surface area of a sphere of radius = surface area of a sphere of radius                                                                  ?(i)                 Step 2: Finding the Potential difference                 The potential difference between two spheres of radii r1 and r2 is given by                                 Since is radial i.e., along r, hence angle between  and  is zero.                 Therefore,                      (Using eqn. (i))                                                                               ?.(ii)                 Step 3: Finding the capacitance C                 Now,             [Using eqn. (ii)]