Spherical capacitor

Under a spherical capacitor or spherical capacitor is understood to mean an electrical capacitor consisting of two concentric , mutually insulated, metallic sphere surfaces.

Spherical capacitor with radii and ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$

For the capacitance of a spherical capacitor with radii and applies ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$

${\displaystyle C=4\pi \varepsilon {\frac {R_{2}R_{1}}{R_{2}-R_{1}}}}$, With ${\displaystyle \varepsilon =\varepsilon _{0}\varepsilon _{r}}$

ε ₀ is the electric field constant . ε _r is the relative permittivity , which is 1 in a vacuum .

Derivation of the formula for the capacity

For an infinitesimally small spherical shell between R ₁ and R ₂ , the known relationship of the plate capacitor applies to the infinitesimally small reciprocal of the capacitance :

${\displaystyle \mathrm {d} {\frac {1}{C}}={\frac {1}{\varepsilon }}\cdot {\frac {\mathrm {d} r}{A(r)}}={\frac {1}{\varepsilon }}\cdot {\frac {\mathrm {d} r}{4\pi r^{2}}}}$

where A ( r ) is the surface of a sphere . If one now integrates, the result is:

${\displaystyle {\frac {1}{C}}=\int \limits _{R_{1}}^{R_{2}}{\frac {1}{\varepsilon }}\cdot {\frac {\mathrm {d} r}{4\pi r^{2}}}={\frac {1}{4\pi \varepsilon }}\cdot \left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)}$

Converted according to the capacity C , this results in the above formula.

Alternatively, the definition can also be used by using the formula in the section Stress between the inner and outer plate . ${\displaystyle C={\frac {Q}{U}}}$

special cases

Very small distance

If so , one can approximately set and receive . ${\displaystyle d=R_{2}-R_{1}\ll R_{1}}$ ${\displaystyle R_{1}=R_{2}=R}$ ${\displaystyle C=4\pi \varepsilon {\frac {R^{2}}{d}}}$

Very large distance

Paper capacitor with a capacity of 5000 cm .

If is, one can approximately set and receive . The capacity is then practically only determined by the radius of the inner sphere. ${\displaystyle R_{1}\ll R_{2}}$ ${\displaystyle R_{2}-R_{1}=R_{2}}$ ${\displaystyle C=4\pi \varepsilon R_{1}}$

This approximation also describes the capacitance of a free-standing ball (also known as a ball electrode), since the counter electrode is very far away here ( and therefore ). The radius of such a spherical electrode in a vacuum was previously used as a unit of measurement for capacitance with the following equivalence: ${\displaystyle R_{2}\to \infty }$ ${\displaystyle R_{1}\ll R_{2}}$

${\displaystyle 1\,\mathrm {cm} \equiv 4\pi \varepsilon _{0}\cdot 1\,\mathrm {cm} \approx 1{,}11265\,\mathrm {pF} }$

Charge and charge density

In the case of the spherical capacitor, it is assumed that the two electrodes are charged with and opposite. These charges are located as surface charges on the inwardly facing spherical surfaces. Then the charge density can be written as , where is Dirac's delta distribution . ${\displaystyle Q}$ ${\displaystyle -Q}$ ${\displaystyle \varrho (r)={\frac {Q}{4\pi R_{1}^{2}}}\,\delta (r-R_{1})-{\frac {Q}{4\pi R_{2}^{2}}}\,\delta (r-R_{2})}$ ${\displaystyle \delta }$

Electric field

The vector of the electric field between the two capacitor shells consists only of the radial component because of the spherical symmetry. This can be calculated with the following formula:

${\displaystyle E(r)={\frac {Q}{4\pi r^{2}\varepsilon _{0}\varepsilon _{r}}}}$ in which ${\displaystyle R_{1}<r<R_{2}}$

The field is not homogeneous, but depends on the distance to the center of the capacitor. There is no electric field inside the electrodes and outside the capacitor. ${\displaystyle r}$

Electrical potential

The electrical potential is a scalar field that only depends on and is calculated as . This integral can be determined in sections: ${\displaystyle r}$ ${\displaystyle \varphi (r)=-\int _{\infty }^{r}E(r')\,dr'}$

• For is . ${\displaystyle r\geq R_{2}}$ ${\displaystyle \varphi (r)=0}$
• For is . ${\displaystyle R_{1}<r<R_{2}}$ ${\displaystyle \varphi (r)=-\int _{R_{2}}^{r}E(r')dr'={\frac {Q}{4\pi \varepsilon _{0}\varepsilon _{r}}}\,\left({\frac {1}{r}}-{\frac {1}{R_{2}}}\right)}$
• For is . ${\displaystyle r\leq R_{1}}$ ${\displaystyle \varphi (r)={\frac {Q}{4\pi \varepsilon _{0}\varepsilon _{r}}}\,\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)}$

Tension between the inner and outer plate

The tension between the inner and outer sphere is calculated as follows:

${\displaystyle U=\varphi (R_{1})-\underbrace {\varphi (R_{2})} _{=0}=\int _{R_{1}}^{R_{2}}E(r)\,\mathrm {d} r\,={\frac {Q}{4\pi \varepsilon _{0}\varepsilon _{r}}}\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)}$

literature

• Eugen Philippow (Ed.): Taschenbuch Elektrotechnik, Volume 1 . Verlag Technik, Berlin 1968, DNB  365695874 , p. 308 ff .
• Klaus Lunze : Introduction to electrical engineering: guidelines and tasks Part I: Electrical circuits with direct current and the electrical field . 3. Edition. Verlag Technik, Berlin 1964, DNB  453110886 , p. 181 ff .

Individual evidence

1. ↑ ^{a b} Eugen Philippow : Fundamentals of electrical engineering . Academic publishing company Geest & Portig K.-G., Leipzig 1967, DNB  457803371 , p. 82 ff .