Answer:
Part(a): The relative capacitance is ![\dfrac{C_{A}}{C_{B}} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BC_%7BA%7D%7D%7BC_%7BB%7D%7D%20%3D%200.33)
Part(b): The relative energy stored is ![\dfrac{U_{A}}{U_{B}} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BU_%7BA%7D%7D%7BU_%7BB%7D%7D%20%3D%200.33)
Part(c): The relative charge stored is ![\dfrac{Q_{A}}{Q_{B}} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BQ_%7BA%7D%7D%7BQ_%7BB%7D%7D%20%3D%200.33)
Explanation:
We know the capacitance (
) of a capacitor having charge (
) and subjected to a potential difference of (
) is given by
![C = \dfrac{Q}{V}](https://tex.z-dn.net/?f=C%20%3D%20%5Cdfrac%7BQ%7D%7BV%7D)
Also, the energy (
) stored by a capacitor can be written as
![U = \dfrac{1}{2}C~V^{2}](https://tex.z-dn.net/?f=U%20%3D%20%5Cdfrac%7B1%7D%7B2%7DC~V%5E%7B2%7D)
Let us assume that the inner radius of the Capacitor B, as shown in the figure, be ![\textbf{r_{i}^{B}}](https://tex.z-dn.net/?f=%5Ctextbf%7Br_%7Bi%7D%5E%7BB%7D%7D)
, the outer radius be
, the inner radius of Capacitor A be
and the outer radius be
.
Given in the problem,
![&& r_{o}^{B} = 2~r_{B}^{i}\\&& r_{o}^{A} = 4~r_{B}^{i}\\&& and~r_{i}^{B} = 4~r_{o}^{B} = 8~r_{B}^{i}](https://tex.z-dn.net/?f=%26%26%20r_%7Bo%7D%5E%7BB%7D%20%3D%202~r_%7BB%7D%5E%7Bi%7D%5C%5C%26%26%20r_%7Bo%7D%5E%7BA%7D%20%3D%204~r_%7BB%7D%5E%7Bi%7D%5C%5C%26%26%20and~r_%7Bi%7D%5E%7BB%7D%20%3D%204~r_%7Bo%7D%5E%7BB%7D%20%3D%208~r_%7BB%7D%5E%7Bi%7D)
Now, the capacitance (
) of a cylindrical capacitor is given by,
![\bf{C = \dfrac{2~\pi~\epsilon_{0}~L}{ln(\dfrac{r_{o}}{r_{i}})}}](https://tex.z-dn.net/?f=%5Cbf%7BC%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7D~L%7D%7Bln%28%5Cdfrac%7Br_%7Bo%7D%7D%7Br_%7Bi%7D%7D%29%7D%7D)
where
is the permittivity of the free space,
is the length of the cylindrical capacitor.
Part(a):
The capacitance of capacitor A,
![C_{A} = \dfrac{2~\pi~\epsilon_{0}L}{ln(\dfrac{r_{o}^{A}}{r_{i}^{A}})} = \dfrac{2~\pi~\epsilon_{0}L}{ln(\dfrac{8~r_{i}^{B}}{r_{i}^{B}})} = \dfrac{2~\pi~\epsilon_{0}L}{ln(8)}](https://tex.z-dn.net/?f=C_%7BA%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%28%5Cdfrac%7Br_%7Bo%7D%5E%7BA%7D%7D%7Br_%7Bi%7D%5E%7BA%7D%7D%29%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%28%5Cdfrac%7B8~r_%7Bi%7D%5E%7BB%7D%7D%7Br_%7Bi%7D%5E%7BB%7D%7D%29%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%288%29%7D)
and the capacitance of capacitor B,
![C_{B} = \dfrac{2~\pi~\epsilon_{0}L}{ln(\dfrac{r_{o}^{B}}{r_{i}^{B}})} = \dfrac{2~\pi~\epsilon_{0}L}{ln(\dfrac{2~r_{i}^{B}}{r_{i}^{B}})} = \dfrac{2~\pi~\epsilon_{0}L}{ln(2)}](https://tex.z-dn.net/?f=C_%7BB%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%28%5Cdfrac%7Br_%7Bo%7D%5E%7BB%7D%7D%7Br_%7Bi%7D%5E%7BB%7D%7D%29%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%28%5Cdfrac%7B2~r_%7Bi%7D%5E%7BB%7D%7D%7Br_%7Bi%7D%5E%7BB%7D%7D%29%7D%20%3D%20%5Cdfrac%7B2~%5Cpi~%5Cepsilon_%7B0%7DL%7D%7Bln%282%29%7D)
giving the relative capacitance of each capacitor to be
![\dfrac{C_{A}}{C_{B}} = \dfrac{ln(2)}{ln(8)} = \dfrac{ln(2)}{3~\ln(2)} = \dfrac{1}{3} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BC_%7BA%7D%7D%7BC_%7BB%7D%7D%20%3D%20%5Cdfrac%7Bln%282%29%7D%7Bln%288%29%7D%20%3D%20%5Cdfrac%7Bln%282%29%7D%7B3~%5Cln%282%29%7D%20%3D%20%5Cdfrac%7B1%7D%7B3%7D%20%3D%200.33)
Part(b):
Energy stored by capacitor A,
![U_{A} = \dfrac{1}{2}~C_{A}~V^{2}](https://tex.z-dn.net/?f=U_%7BA%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D~C_%7BA%7D~V%5E%7B2%7D)
Energy stored by capacitor B,
![U_{B} = \dfrac{1}{2}~C_{B}~V^{2}](https://tex.z-dn.net/?f=U_%7BB%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D~C_%7BB%7D~V%5E%7B2%7D)
giving the relative energy stored by each capacitor to be
![\dfrac{U_{A}}{U_{B}} = \dfrac{C_{A}}{C_{B}} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BU_%7BA%7D%7D%7BU_%7BB%7D%7D%20%3D%20%5Cdfrac%7BC_%7BA%7D%7D%7BC_%7BB%7D%7D%20%3D%200.33)
Part(c):
The charge stored by capacitor A,
![Q_{A} = C_{A}~V](https://tex.z-dn.net/?f=Q_%7BA%7D%20%3D%20C_%7BA%7D~V)
The charge stored by capacitor B,
![Q_{B} = C_{B}~V](https://tex.z-dn.net/?f=Q_%7BB%7D%20%3D%20C_%7BB%7D~V)
giving the relative charge stored by each capacitor to be
![\dfrac{Q_{A}}{Q_{B}} = \dfrac{C_{A}}{C_{B}} = 0.33](https://tex.z-dn.net/?f=%5Cdfrac%7BQ_%7BA%7D%7D%7BQ_%7BB%7D%7D%20%3D%20%20%5Cdfrac%7BC_%7BA%7D%7D%7BC_%7BB%7D%7D%20%3D%200.33)