Cylinder capacitor

An ideal cylinder capacitor is a capacitor that consists of two electrically conductive cylinder jackets between which there is a dielectric ( insulator ). The cylinder jackets are coaxial, have the same height, and the base surfaces of the associated cylinders lie in the same plane.

In the following, electrical field strength in the capacitor, electrical voltage applied between the cylinder casings, electrical charge stored in the capacitor, radius of the inner cylinder casing, radius of the outer cylinder casing, height of the cylinder casing, electrical field constant and relative permittivity of the dielectric. ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle Q}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle l}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \varepsilon _{r}}$

A real cylinder capacitor can consist of two tubes (whose walls are not infinitely thin in contrast to the cylinder jacket), in which case the outer radius of the inner tube and the inner radius of the outer tube. The following formulas apply ideally. Practical applications of the cylindrical capacitor are the Leiden bottle , the tubular capacitor and the coaxial cable . ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$

capacity

Schematic representation of a cylinder capacitor

${\displaystyle C=2\pi \varepsilon _{0}\varepsilon _{r}{\frac {l}{\ln {\frac {R_{2}}{R_{1}}}}}}$

The capacitance can be derived from the electric field as follows:

${\displaystyle C={\frac {Q}{U}}={\frac {Q}{\int {\vec {E}}({\vec {r}})\mathrm {d} {\vec {r}}}}={\frac {Q}{\int \limits _{R_{1}}^{R_{2}}{\frac {Q}{2\pi l\varepsilon _{0}\varepsilon _{r}r}}\mathrm {d} r}}={\frac {2\pi \varepsilon _{0}\varepsilon _{r}l}{\int \limits _{R_{1}}^{R_{2}}{\frac {1}{r}}\mathrm {d} r}}=2\pi \varepsilon _{0}\varepsilon _{r}{\frac {l}{\ln {\frac {R_{2}}{R_{1}}}}}}$

Electric field

The field between the cylinder jackets is not homogeneous, but decreases radially. It can be derived according to Gaussian law . To do this, choose the closed surface of a cylinder with a radius . The unit vector points in the radial direction of the cylinder coordinates . ${\displaystyle \quad R_{1}<r<R_{2}}$ ${\displaystyle {\vec {\mathrm {e} _{r}}}}$

${\displaystyle {\frac {Q}{\varepsilon }}=\oint _{A}{\vec {E}}\;\cdot \mathrm {d} {\vec {A}}=\oint _{A}E{\vec {\mathrm {e} _{r}}}\cdot \mathrm {d} {\vec {A}}=\oint _{A}E\mathrm {d} A=\int _{0}^{l}\int _{0}^{2\pi }Er\mathrm {d} \varphi \mathrm {d} z=E2\pi lr}$

${\displaystyle E(r)={\frac {Q}{2\pi rl\varepsilon }}\quad ,\quad \varepsilon =\varepsilon _{0}\varepsilon _{r}}$

The mean electric field strength corresponds to that of a plate capacitor.

${\displaystyle {\bar {E}}={\frac {1}{R_{2}-R_{1}}}\int _{R_{1}}^{R_{2}}E(r)\,\mathrm {d} r={\frac {U}{R_{2}-R_{1}}}}$

In the ideal case, there is no electrical field caused by the capacitor outside.

tension

between inner and outer cylinder jacket :

${\displaystyle U=\int _{R1}^{R2}{\vec {E}}\;(r)\cdot \mathrm {d} {\vec {r}}=\int _{R1}^{R2}E(r){\vec {e}}_{r}\cdot \mathrm {d} {\vec {r}}=\int _{R1}^{R2}E(r)\mathrm {d} r=\int _{R1}^{R2}{\frac {Q}{2\pi rl\varepsilon }}\mathrm {d} r={\frac {Q}{2\pi l\varepsilon }}\cdot \ln {\frac {R_{2}}{R_{1}}}}$

With

${\displaystyle \varepsilon =\varepsilon _{0}\cdot \varepsilon _{r}}$

literature

• Douglas C. Giancoli: Physics . 3. Edition. Pearson Studies, ISBN 978-3-86894-023-7 , pp. 822 ff .