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

A point dipole p is imbedded at the center of a sphere of linear dielectric material (with radius R and dielectric constant \epsilon_{r}). Find the electric potential inside and outside the sphere.

\left[\text { Answer: } \frac{p \cos \theta}{4 \pi \epsilon r^{2}}\left(1+2 \frac{r^{3}}{R^{3}} \frac{\left(\epsilon_{r}-1\right)}{\left(\epsilon_{r}+2\right)}\right),(r \leq R) ; \frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}}\left(\frac{3}{\epsilon_{r}+2}\right),(r \geq R)\right]
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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)

\left\{\begin{array}{l}\text { Outside: } V(r, \theta)=\sum_{l=0}^{\infty} \frac{B_{l}}{r^{l+1}} P_{l}(\cos \theta)(Eq. 3.72) \\\text { Inside: } \quad V(r, \theta)=\frac{1}{4 \pi \epsilon_{0}} \frac{p \cos \theta}{\epsilon_{r} r^{2}}+\sum_{l=0}^{\infty} A_{l} r^{l} P_{l}(\cos \theta)(\text { Eqs. } 3.66,3.102)\end{array}\right\}

 

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)

V \text { continuous at } R \Rightarrow\left\{\begin{array}{ll}\frac{B_{l}}{R^{l+1}}=A_{l} R^{l}, & \text { or } B_{l}=R^{2 l+1} A_{l}(l \neq 1) \\\frac{B_{1}}{R^{2}}=\frac{1}{4 \pi \epsilon_{0}} \frac{p}{\epsilon_{r} R^{2}}+A_{1} R, & \text { or } \quad B_{1}=\frac{p}{4 \pi \epsilon_{0} \epsilon_{r}}+A_{1} R^{3}\end{array}\right\}

 

\left.\frac{\partial V}{\partial r}\right|_{R+}-\left.\frac{\partial V}{\partial r}\right|_{R-}=-\sum(l+1) \frac{B_{l}}{R^{l+2}} P_{l}(\cos \theta)+\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)=-\frac{1}{\epsilon_{0}} \sigma_{b}

 

=-\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) .

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