electrodynamics:res2.32-1 [Задачи по физике]

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$$
\frac{\partial \varphi }{\partial r}
=
\frac{\partial}{\partial r}\left.\left(\frac{q}{\sqrt{r^{2}+b^{2}-2rb\cdot\cos(\theta)}}-\frac{q}{\sqrt{r^{2}+b^{2}+2rb\cdot\cos(\theta)}}+\right.\right.
$$
$$
\left.\left.\frac{-q\frac{a}{b}}{\sqrt{r^{2}+(\frac{a^{2}}{b})^{2}-2r\frac{a^{2}}{b}\cdot\cos(\theta)}}-\frac{-q\frac{a}{b}}{\sqrt{r^{2}+(\frac{a^{2}}{b})^{2}+2r\frac{a^{2}}{b}\cdot\cos(\theta)}}\right)\right|_{r=a}=
$$
$$
\left.q\left(\frac{r-b\cdot\cos(\theta)}{\left(r^{2}+b^{2}-2rb\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{r+b\cdot\cos(\theta)}{\left(r^{2}+b^{2}+2rb\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\right.\right.
$$
$$
\left.\left.\frac{\frac{a}{b}\left(r-\frac{a^{2}}{b}\cdot\cos(\theta)\right)}{\left(r^{2}+(\frac{a^{2}}{b})^{2}-2r\frac{a^{2}}{b}\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\frac{a}{b}\left(r+\frac{a^{2}}{b}\cdot\cos(\theta)\right)}{\left(r^{2}+(\frac{a^{2}}{b})^{2}+2r\frac{a^{2}}{b}\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)\right|_{r=a}=
$$
$$
q\left(\frac{a-b\cdot\cos(\theta)}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{a+b\cdot\cos(\theta)}{\left(a^{2}+b^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-
\right.
$$
$$
\left.\frac{\left(\frac{a}{b}\right)^{2}\left(b-a\cdot\cos(\theta)\right)}{\left(\frac{a}{b}\right)^{3}\left(b^{2}+a^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\left(\frac{a}{b}\right)^{2}\left(b+a\cdot\cos(\theta)\right)}{\left(\frac{a}{b}\right)^{3}\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)=
$$
$$
q\left(\frac{a-b\cdot\cos(\theta)}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{\frac{b}{a}\left(b-a\cdot\cos(\theta)\right)}{\left(b^{2}+a^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\right.
$$
$$
\left.\frac{a+b\cdot\cos(\theta)}{\left(a^{2}+b^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\frac{b}{a}\left(b+a\cdot\cos(\theta)\right)}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)=
$$
$$
q\left(\frac{a^{2}-b^{2}}{a\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{b^{2}-a^{2}}{a\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)=
$$
$$
q\frac{b^{2}-a^{2}}{a}\left(\frac{1}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{1}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right).
$$