Spread the love

Cylindrical Capacitor | Capacitance Of Cylindrical Capacitor

Cylindrical Capacitor | Capacitance Of Cylindrical Capacitor :- A cylindrical capacitor is a type of capacitor in which two coaxial cylindrical conductors are present, and the space between them is filled with a dielectric material (air or any other insulating material).

Structure of a Cylindrical Capacitor :

Inner cylinder: The inner cylinder is a solid or hollow cylindrical rod.
Outer cylinder: The outer cylinder is a hollow cylindrical shell surrounding the inner cylinder and is grounded (earthed).
Dielectric medium: The space between the two cylinders is filled with a dielectric material, which may be air, vacuum, or any other insulating (non-conducting) substance.

Working Principle of a Cylindrical Capacitor :

Let the radii of the inner and outer cylinders and be r₁ and r₂ respectively, and their length be . The inner cylinder is given a charge +Q, due to which a charge -Q is induced on the inner surface of cylinder and a charge +Q on its outer surface. Now, since the outer surface of cylinder $B$ is earthed, the charge flows into the Earth.

The electric field lines originate from the outer surface of cylinder A and terminate on the inner surface of cylinder B. The electric energy is stored in the region between the two cylinders. Inside cylinder A and outside cylinder B, the electric field (and hence the electric field lines) is zero. To ensure that the electric field in the region between the cylinders is uniform, we assume that the length of the cylinders L >> (r₁and r₂).

Capacitance of a Cylindrical Capacitor

Linear charge density,

$\displaystyle \lambda \text{=}\frac{Q}{L}$

The electric field between the cylinders,

$\displaystyle E=\frac{\lambda }{2\pi {{\varepsilon }_{0}}r}=\frac{{}^{Q}\!\!\diagup\!\!{}_{L}\;}{2\pi {{\varepsilon }_{0}}r}=\frac{Q}{2\pi {{\varepsilon }_{0}}Lr}$

Potential difference between the cylinders,

$\displaystyle V=\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\frac{Q}{2\pi {{\varepsilon }_{0}}Lr}dr}=\frac{Q}{2\pi {{\varepsilon }_{0}}L}\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\frac{1}{r}dr}$

$\displaystyle \Rightarrow V=\frac{Q}{2\pi {{\varepsilon }_{0}}L}\left[ \ln r \right]_{{{r}_{1}}}^{{{r}_{2}}}$

$\displaystyle V=\frac{Q}{2\pi {{\varepsilon }_{0}}L}ln\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)$

Therefore, the capacitance of the cylindrical capacitor is,

$\displaystyle C=\frac{Q}{V}=\frac{Q}{\frac{Q}{2\pi {{\varepsilon }_{0}}L}ln\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)}$

$\displaystyle C=\frac{2\pi {{\varepsilon }_{0}}L}{lo{{g}_{e}}\left( {}^{{{r}_{2}}}\!\!\diagup\!\!{}_{{{r}_{1}}}\; \right)}$ …..(1)

Equation (1) is the expression for the capacitance of a cylindrical capacitor.

If a material with dielectric constant ε_r is filled between the cylinders, then the capacitance of the cylindrical capacitor,

$\displaystyle {{C}_{m}}={{\varepsilon }_{r}}C=\frac{2\pi {{\varepsilon }_{0}}{{\varepsilon }_{r}}L}{lo{{g}_{e}}\left( {}^{{{r}_{2}}}\!\!\diagup\!\!{}_{{{r}_{1}}}\; \right)}$ …..(2)

Cylindrical Capacitor | Capacitance Of Cylindrical Capacitor

Structure of a Cylindrical Capacitor :

Working Principle of a Cylindrical Capacitor :

Capacitance of a Cylindrical Capacitor

Like this: