Capacity of Various Capacitor
Category : JEE Main & Advanced
(1) Parallel plate capacitor : It consists of two parallel metallic plates (may be circular, rectangular, square) separated by a small distance. If A = Effective overlapping area of each plate.
(i) Electric field between the plates : \[E=\frac{\sigma }{{{\varepsilon }_{0}}}=\frac{Q}{A{{\varepsilon }_{0}}}\]
(ii) Potential difference between the plates : \[V=E\times d=\frac{\sigma \,d}{{{\varepsilon }_{0}}}\]
(iii) Capacitance : \[C=\frac{{{\varepsilon }_{\mathbf{0}}}A}{d}\]. In C.G.S. : \[C=\frac{A}{\mathbf{4}\pi d}\]
(iv) If a dielectric medium of dielectric constant K is filled completely between the plates then capacitance increases by K times i.e. \[C'=\frac{K{{\varepsilon }_{0}}A}{d}\] \[\Rightarrow \]\[C'=KC\]
(v) The capacitance of parallel plate capacitor depends on \[A(C\,\propto A)\] and \[d\,\left( C\propto \frac{1}{d} \right)\]. It does not depend on the charge on the plates or the potential difference between the plates.
(vi) If a dielectric slab is partially filled between the plates
\[\Rightarrow \] \[C'=\frac{{{\varepsilon }_{0}}A}{d-t+\frac{t}{K}}\]
(vii) If a number of dielectric slabs are inserted between the plate as shown
\[C'=\frac{{{\varepsilon }_{0}}A}{d-({{t}_{1}}+{{t}_{2}}+{{t}_{3}}+........)+\left( \frac{{{t}_{1}}}{{{K}_{1}}}+\frac{{{t}_{2}}}{{{K}_{2}}}+\frac{{{t}_{3}}}{{{K}_{3}}}+........ \right)}\]
(viii) When a metallic slab is inserted between the plates \[C'=\frac{{{\varepsilon }_{0}}A}{(d-t)}\]
If metallic slab fills the complete space between the plates (i.e. \[t=d\]) or both plates are joined through a metallic wire then capacitance becomes infinite.
(ix) Force between the plates of a parallel plate capacitor.
\[|F|\,=\frac{{{\sigma }^{2}}A}{2{{\varepsilon }_{0}}}=\frac{{{Q}^{2}}}{2{{\varepsilon }_{0}}A}=\frac{C{{V}^{2}}}{2d}\]
(x) Energy density between the plates of a parallel plate capacitor.
Energy density \[=\frac{Energy}{Volume}\]\[=\frac{1}{2}\,{{\varepsilon }_{0}}{{E}^{2}}.\]
Variation of different variable (Q, C, V, E and U) of parallel plate capacitor
| Quantity |
Battery is Removed
|
Battery Remains connected
|
| Capacity | \[C'=KC\] | \[C'=KC\] |
| Charge | \[Q'=Q\] | \[Q'=KQ\] |
| Potential | \[V'=V/K\] | \[V'=V\] |
| Intensity | \[E'=E/K\] | \[E'=E\] |
| Energy | \[U'=U/K\] | \[U'=KU\] |
(2) Spherical capacitor : It consists of two concentric conducting spheres of radii \[a\] and \[b\]\[(a<b)\]. Inner sphere is given charge \[+Q,\] while outer sphere is earthed
(i) Potential difference : Between the spheres is
\[V=\frac{Q}{4\pi {{\varepsilon }_{0}}a}-\frac{Q}{4\pi {{\varepsilon }_{0}}b}\]
(ii) Capacitance : \[C=4\pi {{\varepsilon }_{0}}.\frac{ab}{b-a}\].
In C.G.S. \[C=\frac{ab}{b-a}\]. In the presence of dielectric medium (dielectric constant K) between the spheres \[C'=4\pi {{\varepsilon }_{0}}K\frac{ab}{b-a}\]
(iii) If outer sphere is given a charge \[+Q\] while inner sphere is earthed
Induced charge on the inner sphere
\[Q'=-\frac{a}{b}.Q\] and capacitance of
the system \[C'=4\pi {{\varepsilon }_{0}}.\frac{{{b}^{2}}}{b-a}\]
This arrangement is not a capacitor. But it?s capacitance is equivalent to the sum of capacitance of spherical capacitor and spherical conductor i.e. \[4\pi {{\varepsilon }_{0}}.\frac{{{b}^{2}}}{b-a}=4\pi {{\varepsilon }_{0}}\frac{ab}{b-a}+4\pi {{\varepsilon }_{0}}b\]
(3) Cylindrical capacitor : It consists of two concentric cylinders of radii \[a\] and \[b\]\[(a<b)\], inner cylinder is given charge \[+Q\] while outer cylinder is earthed. Common length of the cylinders is l then
\[C=\frac{2\pi {{\varepsilon }_{0}}l}{{{\log }_{e}}\left( \frac{b}{a} \right)}\]
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