Question 4.37: A point dipole p is imbedded at the center of a sphere of li...

In view of Eq. 4.39, the net dipole moment at the center is  p ^{\prime}= p -\frac{\chi_{e}}{1+\chi_{e}} p =\frac{1}{1+\chi_{e}} p =\frac{1}{\epsilon_{r}} p potential produced by  p ^{\prime} \text { (at the center) and } \sigma_{b} (at R). Use separation of variables:

\rho_{b}=-\nabla \cdot P =-\nabla \cdot\left(\epsilon_{0} \frac{\chi_{e}}{\epsilon} D \right)=-\left(\frac{\chi_{e}}{1+\chi_{e}}\right) \rho_{f}                                (4.39)

V(r, \theta)=\sum_{l=0}^{\infty} \frac{B_{l}}{r^{l+1}} P_{l}(\cos \theta)                          (3.72)

V(r, \theta)=\sum_{l=0}^{\infty} A_{l} r^{l} P_{l}(\cos \theta)                            (3.66)

V_{ dip }(r, \theta)=\frac{\hat{ r } \cdot p }{4 \pi \epsilon_{0} r^{2}}=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}}                             (3.102)

=-\frac{1}{\epsilon_{0}} P \cdot \hat{ r }=-\frac{1}{\epsilon_{0}}\left(\epsilon_{0} \chi_{e} E \cdot \hat{ r }\right)=\left.\chi_{e} \frac{\partial V}{\partial r}\right|_{R-}=\chi_{e}\left\{-\frac{1}{4 \pi \epsilon_{0}} \frac{2 p \cos \theta}{\epsilon_{r} R^{3}}+\sum l A_{l} R^{l-1} P_{l}(\cos \theta)\right\} .

-(l+1) \frac{B_{l}}{R^{l+2}}-l A_{l} R^{l-1}=\chi_{e} l A_{l} R^{l-1}(l \neq 1) ; \text { or }-(2 l+1) A_{l} R^{l-1}=\chi_{e} l A_{l} R^{l-1} \Rightarrow A_{l}=0(\ell \neq 1) .

\text { For } l=1:-2 \frac{B_{1}}{R^{3}}+\frac{1}{4 \pi \epsilon_{0}} \frac{2 p}{\epsilon_{r} R^{3}}-A_{1}=\chi_{e}\left(-\frac{1}{4 \pi \epsilon_{0}} \frac{2 p}{\epsilon_{r} R^{3}}+A_{1}\right)-B_{1}+\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}}-\frac{A_{1} R^{3}}{2}=-\frac{1}{4 \pi \epsilon_{0}} \frac{\chi_{e} p}{\epsilon_{r}}+\chi_{e} \frac{A_{1} R^{3}}{2} ;

-\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}}-A_{1} R^{3}+\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}}-\frac{A_{1} R^{3}}{2}=-\frac{1}{4 \pi \epsilon_{0}} \frac{\chi_{e} p}{\epsilon_{r}}+\chi_{e} \frac{A_{1} R^{3}}{2} \Rightarrow \frac{A_{1} R^{3}}{2}\left(3+\chi_{e}\right)=\frac{1}{4 \pi \epsilon_{0}} \frac{\chi_{e} p}{\epsilon_{r}} .

\Rightarrow A_{1}=\frac{1}{4 \pi \epsilon_{0}} \frac{2 \chi_{e} p}{R^{3} \epsilon_{r}\left(3+\chi_{e}\right)}=\frac{1}{4 \pi \epsilon_{0}} \frac{2\left(\epsilon_{r}-1\right) p}{R^{3} \epsilon_{r}\left(\epsilon_{r}+2\right)} ; \quad B_{1}=\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}}\left[1+\frac{2\left(\epsilon_{r}-1\right)}{\left(\epsilon_{r}+2\right)}\right]=\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}} \frac{3 \epsilon_{r}}{\epsilon_{r}+2} .

V(r, \theta)=\left(\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}}\right)\left(\frac{3}{\epsilon_{r}+2}\right)(r \geq R) .

Meanwhile, for  r \leq R, V(r, \theta)=\frac{1}{4 \pi \epsilon_{0}} \frac{p \cos \theta}{\epsilon_{r} r^{2}}+\frac{1}{4 \pi \epsilon_{0}} \frac{p r \cos \theta}{R^{3}} \frac{2\left(\epsilon_{r}-1\right)}{\epsilon_{r}\left(\epsilon_{r}+2\right)}

=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2} \epsilon_{r}}\left[1+2\left(\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\right) \frac{r^{3}}{R^{3}}\right](r \leq R) .