Research website of Vyacheslav Gorchilin
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The potential of a charged sphere. Values of certain functions
In the previous section we have derived formula (19) for the distribution of electric potential along the radius of the ball, depending on the distribution of volume charge density. This formula made it possible to find physical meaning of this phenomenon and simplified mathematical calculations. In this section we find a formula for some popular functions for both absolute capacity, and the difference between the ball surface and the layer at a known depth.
The output for each function will be divided into three parts. In the first we derive a General formula for the potential distribution along the radius in dependence on the same distribution of volume charge density. In the second part we find the potential difference between the ball surface and the layer located at some depth \(h\). The third part will be dedicated to the simplification of the formula from the second section, under the assumption that the depth \(h\) is at least several orders of magnitude smaller than the overall radius of the ball \(R\). This calculation is based on the decomposition of functions in power series the Taylor-Maclaurin [1], accurate to the second member.
Dependence: \(\rho(r) = \rho_0\, (r/R)^n\,\) where \(n \ge -1\)
1. \[\varphi(r) = { \rho_0\, R^{2} \over \varepsilon_0} {n + 3 - (r/R)^{n+2} \over (n+2) (n+3)} \qquad (1.1)\] 2. \[\Delta \varphi(h) = { \rho_0\, R^{2} \over \varepsilon_0} {(1- \frac{h}{R})^{n+2} - 1 \over (n+2) (n+3)} \qquad (1.2)\] 3. \[\Delta \varphi(h) \approx { \rho_0\, R\, h \over \varepsilon_0 (n+3)} \left({n+1 \over 2} \frac{h}{R} - 1 \right), \quad R \gg h \qquad (1.3)\]

Dependence: \(\rho(r) = \rho_0\, e^{\alpha r}\)
1. \[\varphi(r) = { \rho_0 \over \varepsilon_0\, \alpha^{2}} \left[ \frac{2}{\alpha r} (e^{\alpha r} - 1) + e^{\alpha R} (\alpha R - 1) - e^{\alpha r} \right] \qquad (2.1)\] 2. \[\Delta \varphi(h) = {\rho_0 \over \varepsilon_0\, \alpha^{2}} \left[{2 e^{\alpha R} \over \alpha (R-h)} \left(\frac{R-h}{R} - e^{-\alpha h} \right) - e^{\alpha R} \left( 1-e^{-\alpha h} \right) + {2 h \over \alpha R(R-h)} \right] \qquad (2.2)\] 3. \[\Delta \varphi(h) \approx {\rho_0\, h \over \varepsilon_0\, \alpha} \left[\frac{(1 + \delta)(\alpha R - 2) e^{\alpha R}}{\alpha R} + {2 (1 + \delta)(e^{\alpha R} - 1) \over \alpha^2 R^2} - \frac{\alpha h e^{\alpha R}}{2} \right] \qquad (2.3)\] this is a reduction: \(\delta = h/R\). This formula is simplified if there are some additional conditions: \[\Delta \varphi(h) \approx {\rho_0\, h\, R \over \varepsilon_0} \left[e-2 + \frac{h}{R} {e-4 \over 2} \right], \quad \alpha R = 1 \qquad (2.3.1)\] \[\Delta \varphi(h) \approx -{\rho_0\, h\, R \over \varepsilon_0} \left[{1+2e(e-1) \over e} + \frac{h}{R}{11-4e \over 2e} \right], \quad \alpha, R = -1 \qquad (2.3.2)\]
The materials used
  1. Wikipedia. A series of Taylor and Maclaurin.