Cylinderical capacitor

Cylinderical capacitor

• A cylinderical capacitor is made up of a conducting cylinder or wire of radius a surrounded by another concentric cylinderical shell of radius b (b>a).
  
  
• Let L be the length of both the cylinders and charge on inner cylender is +Q and charge on outer cylinder is -Q.
  
• For calculate electric field between the conductors using Gauss's law consider a Gaussian surface of radius r and length L¹ as shown in figure 4.
  
  
  
  
• According to Gauss's law flux through this surface is $ \frac {q}{\epsilon _0}$ where q is net charge inside this surface.
  
• We know that electric flux is given by
  $\phi =E.A$
  $=EA cos \theta$
  $=EA$
  since electric field is constant in magnitude on the Gaussian surface and is perpendicular to this surface. Thus,
  $\phi=E(2 \pi rL)$
  since $\phi=\frac {q}{\epsilon _0}$
  => $ E(2 \pi rL)=\frac {\lambda L}{\epsilon _0}$
  where $\lambda = Q/L$ = charge per unit length
  So, $E=\frac{\lambda }{2\pi \epsilon _{0}r}$
  
• If potential at inner cylinder is V_a and V_b is potential of outer cylinder then potential difference between both the cylinders is
  V=V_a and V_b=∫Edr
  where limits of integration goes from a to b.
  
  
  
• Potential of inner conductor is greater then that of outer conductor because inner cylinder carries positive charge. Thus potential difference is $$V=\frac {Qln(b/a)}{2\pi \epsilon _{0}L}$$
  
• Thus capacitance of cylinderical capacitor is
  $C=\frac {Q}{V}
  or,
  $C=\frac{2\pi \epsilon _{0}L}{ln (b/a)}$ ------(5)
  
• From equation 5 it can easily be concluded that capacitance of a cylinderical capacitor depends on length of cylinders.
  
• More is the length of cylinders , more charge could be stored on the capacitor for a given potential difference.
  

• Notes
  • Capacitance and Capacitor
  • Parallel plate capacitor
  • Cylinderical capacitor
  • Spherical capacitor
  • Capacitors in series and parallel combinations
  • Energy stored in a capacitor
  • Effect of Dielectric on Capacitance
• Assignment
  • Capacitance MCQ
  • Capacitance Numericals