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 byC=4ϵ∏r1r2/r1-r2where r1 and r2 are the radii of outer and inner spheres,respectively.
The given figure is shown below.
The potential difference between the two shells is given as,
V = Q 4 π ε 0 r 2 − Q 4 π ε 0 r 1
Where, the radius of outer shell is r 1 , the radius of the inner shell is r 2 , the charge of the inner surface of the outer shell is + Q , the charge of the outer surface of the inner shell is − Q and the permittivity of free space is ε 0 .
By simplifying the above expression, we get
V = Q 4 π ε 0 [ 1 r 2 − 1 r 1 ] = Q ( r 1 − r 2 ) 4 π ε 0 r 1 r 2
The capacitance of the system is given as,
C = Q V
By substituting the given value in the above expression, we get
C = Q Q ( r 1 − r 2 ) 4 π ε 0 r 1 r 2 = 4 π ε 0 r 1 r 2 ( r 1 − r 2 )
Thus, the capacitance of a spherical capacitor is 4 π ε 0 r 1 r 2 ( r 1 − r 2 ) .
The given figure is shown below.
The potential difference between the two shells is given as,
Where, the radius of outer shell is
By simplifying the above expression, we get
The capacitance of the system is given as,
By substituting the given value in the above expression, we get
Thus, the capacitance of a spherical capacitor is
