A sperical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports. Show that the capacitance of this spherical capacitor is given by `C = (4pi in_(0) r_(1) r_(2))/(r_(1) - r_(2))`,
Where `r_(1) and r_(2)` are radill of outer and inner spheres respectively.

Options

A. `(4piepsi_(0)r_(1)r_(2))/(r_(1)-r_(2))`
B. `(4piepsi_(0)-(r_(2)-r_(1)))/(r_(1)r_(2))`
C. `(r_(1)r_(2))/(4 pi epsi_(0)(r_(2)-r_(1)))`
D. `((r_(1)-r_(2)))/(4 pi epsi_(0)r_(1)r_(2))`

Answer

A

Explanation

As show in figure , +q charge spreads uniformly on inner surface of outer sphere of radius `r_(1)`. The iduced charge ` -q` spreads uniformly on the outer surface of inner sphere of radius `r_(2)` .
The outer surface of outer sphere is earthed. Due to electrostatic shielding .
`E = 0` for ` r lt r_(2)` and `E = 0` for `r lt r_(1)` .
In the space between the two sphere, Potential difference between two spheres,
`V = (q)/(4piepsi_(0)r_(2)) - (q)/(4pi epsi_(0) r_(1)) =(q)/(4piepsi_(0))[(1)/(r_(2))-(1)/(r_(1))]`
`V = (q)/(4 piepsi_(0))((r_(1)-r_(2))/(r_(1)r_(2)))`
Also `c = (q)/(V)`
`therefore C = (4 pi epsi_(0)r_(1)r_(2))/(r_(1)-r_(2))" "("using (i)")`