Problem 3.22 A spherical shell of radius $R$ carries a uniform suface charge $\sigma_{0}$ on the "*north- ern" hemisphere and a uniform surface charge $- \sigma_{0}$ on the "southern" hemisphere. Find the ptentialinsid nd outside the per, alulating the coeficients expityupto $A_{6}$ and $B_{6}$.

Options:
A. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {2 e_{0}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {2} \left( \frac{r} {R} \right)^{5} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {5} \left( \frac{r} {R} \right)^{7} P_{2} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{2}} {2 e_{0} r^{5}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {2} \left( \frac{R} {r} \right)^{5} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {5} \left( \frac{R} {r} \right)^{7} P_{2} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
B. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {3 e_{0}} \left[ P_{3} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{r} {R} \right)^{3} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {7} \left( \frac{r} {R} \right)^{5} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{3}} {2 e_{0} r^{2}} \left[ P_{2} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{R} {r} \right)^{3} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {7} \left( \frac{R} {r} \right)^{5} P_{3} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
C. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {2 e_{0}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {4} \left( \frac{r} {R} \right)^{2} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{r} {R} \right)^{4} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{2}} {2 e_{0} r^{2}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {4} \left( \frac{R} {r} \right)^{2} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{R} {r} \right)^{4} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
D. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {4 e_{0}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {6} \left( \frac{r} {R} \right)^{4} P_{6} ( \operatorname{c o s} \theta)+\frac{1} {10} \left( \frac{r} {R} \right)^{3} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{4}} {3 e_{0} r^{2}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {6} \left( \frac{R} {r} \right)^{4} P_{6} ( \operatorname{c o s} \theta)+\frac{1} {10} \left( \frac{R} {r} \right)^{3} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
E. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {7 e_{0}} \left[ P_{5} ( \operatorname{c o s} \theta)-\frac{1} {3} \left( \frac{r} {R} \right)^{2} P_{1} ( \operatorname{c o s} \theta)+\frac{1} {6} \left( \frac{r} {R} \right)^{6} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{7}} {2 e_{0} r^{2}} \left[ P_{5} ( \operatorname{c o s} \theta)-\frac{1} {3} \left( \frac{R} {r} \right)^{2} P_{1} ( \operatorname{c o s} \theta)+\frac{1} {6} \left( \frac{R} {r} \right)^{6} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
F. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {2 e_{0}} \left[ P_{4} ( \operatorname{c o s} \theta)-\frac{1} {4} \left( \frac{r} {R} \right)^{6} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{r} {R} \right)^{2} P_{7} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{2}} {5 e_{0} r^{2}} \left[ P_{4} ( \operatorname{c o s} \theta)-\frac{1} {4} \left( \frac{R} {r} \right)^{6} P_{3} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{R} {r} \right)^{2} P_{7} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
G. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {3 e_{0}} \left[ P_{4} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{r} {R} \right)^{3} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{r} {R} \right)^{6} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{3}} {4 e_{0} r^{2}} \left[ P_{4} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{R} {r} \right)^{3} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {8} \left( \frac{R} {r} \right)^{6} P_{5} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
H. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {2 e_{0}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {6} \left( \frac{r} {R} \right)^{4} P_{6} ( \operatorname{c o s} \theta)+\frac{1} {10} \left( \frac{r} {R} \right)^{2} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{2}} {2 e_{0} r^{4}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {6} \left( \frac{R} {r} \right)^{4} P_{6} ( \operatorname{c o s} \theta)+\frac{1} {10} \left( \frac{R} {r} \right)^{2} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
I. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {2 e_{0}} \left[ P_{2} ( \operatorname{c o s} \theta)-\frac{1} {3} \left( \frac{r} {R} \right)^{3} P_{4} ( \operatorname{c o s} \theta)+\frac{1} {9} \left( \frac{r} {R} \right)^{5} P_{6} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{2}} {2 e_{0} r} \left[ P_{2} ( \operatorname{c o s} \theta)-\frac{1} {3} \left( \frac{R} {r} \right)^{3} P_{4} ( \operatorname{c o s} \theta)+\frac{1} {9} \left( \frac{R} {r} \right)^{5} P_{6} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
J. $$
\left\{\begin{array} {l} {{{\frac{\sigma_{0} r} {6 e_{0}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{r} {R} \right)^{5} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {7} \left( \frac{r} {R} \right)^{3} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \qquad( r \leq R ),}}} \\ {{{\frac{\sigma_{0} R^{6}} {4 e_{0} r^{2}} \left[ P_{1} ( \operatorname{c o s} \theta)-\frac{1} {5} \left( \frac{R} {r} \right)^{5} P_{2} ( \operatorname{c o s} \theta)+\frac{1} {7} \left( \frac{R} {r} \right)^{3} P_{4} ( \operatorname{c o s} \theta)+\ldots\right], \ ( r \geq R ).}}} \\ \end{array} \right\}
$$
Answer: C