physicscatalyst.com logo




Spherical capacitor





Spherical capacitor

  • A spherical capacitor consists of a solid or hollow spherical conductor of radius a , surrounded by another hollow concentric spherical of radius b shown below in figure 5
    Spherical capacitor

  • Let +Q be the charge given to the inner sphere and -Q be the charge given to the outer sphere.
  • The field at any point between conductors is same as that of point charge Q at the origin and charge on outer shell does not contribute to the field inside it.
  • Thus electric field between conductors is $$E=\frac{Q}{2\pi \epsilon _{0}r^{2}}$$
  • Potential difference between two conductors is
    $V=V_a -V_b$
    $=- \int E.dr $
    where limits of integration goes from a to b.
    On integrating we get potential difference between to conductors as
    $$V=\frac{Q(b-a)}{4\pi \epsilon _{0}ba}$$
  • Now , capacitance of spherical conductor is
    $C= \frac {Q}{V} $
    or,
    $C=\frac{4\pi \epsilon _{0}ba}{(b-a)}$ ----(1)
  • again if radius of outer conductor approaches to infinity then from equation 6 we have
    $C=4 \pi \epsilon _{0} a$ ----(2)
  • Equation 2 gives the capacitance of single isolated sphere of radius a.
  • Thus capacitance of isolated spherical conductor is proportional to its radius.



Spherical capacitor when inner sphere is earthed

  • If a positive charge of Q coulombs is given to the outer sphere B, it will distribute itself over both its inner and outer surfaces.
  • Let the charges of $Q_1$ and $Q_2$ coulombs be at the inner and outer surfaces respectively of sphere B where $Q = Q_1 +Q_2$,
  • The charge + $Q_1$ on the inner surface of outer sphere B will induce a charge of -$Q_1$ coulombs on the outer surface of inner sphere A and + $Q_1$ coulombs on the inner surface of sphere A, which will go to earth.
  • Now there are two capacitors connected in parallel.
    (i) One capacitor consists outer surface of sphere B and earth having capacitance $C_1 = 4 \pi \epsilon _0 b$ farads
    (ii) Second capacitor consisting of inner surface of outer sphere B and the outer surface of inner sphere A having capacitance
    $C_2=\frac{4\pi \epsilon _{0}ba}{(b-a)}$
  • Final Capacitance
    $C=C_1+C_2=4 \pi \epsilon _0 b + \frac{4\pi \epsilon _{0}ba}{(b-a)}= \frac{4\pi \epsilon _{0}b^2}{(b-a)}$
link to this page by copying the following text





Go back to Class 12 Main Page using below links
Class 12 Maths Class 12 Physics Class 12 Chemistry Class 12 Biology





Note to our visitors :-

Thanks for visiting our website.
DISCLOSURE: THIS PAGE MAY CONTAIN AFFILIATE LINKS, MEANING I GET A COMMISSION IF YOU DECIDE TO MAKE A PURCHASE THROUGH MY LINKS, AT NO COST TO YOU. PLEASE READ MY DISCLOSURE FOR MORE INFO.