The magnetic dipole $m_{\circ}$ located at a distance $d$ ($d > a$) from the center of a superconducting sphere with radius $a$, which is in an ideal Meissner state, is oriented along the radial direction of the sphere. Prove that the image dipole of $m$ is $\boldsymbol{m}^{\prime} = -\left( a / d \right)^{3} \boldsymbol{m}$, positioned inside the sphere at $z = a^{2} / d$.

Options:
A. $$
\frac{m ( r \operatorname{c o s} \phi-d )} {\left[ r^{3}+d^{2}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \phi-\left( \frac{a} {d} \right)^{3} d \right)} {\left[ a^{2}+\left( \frac{r} {a} \right)^{4} d^{2}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}
$$
B. $$
\frac{m ( r \operatorname{c o s} \theta+a )} {\left[ r^{2}+d^{3}-2 r a \operatorname{c o s} \theta\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \theta-\left( \frac{a} {d} \right)^{3} d a \right)} {\left[ a^{2}+\left( \frac{r} {a} \right)^{2} d^{3}-2 a r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}
$$
C. $$
\frac{m ( r \operatorname{c o s} \theta-d )} {\left[ r^{2}+d^{2}-2 r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \theta-\left( \frac{a} {d} \right)^{2} d \right)} {\left[ a^{2}+\left( \frac{r} {a} \right)^{2} d^{2}-2 r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}
$$
D. $$
\frac{m ( r \operatorname{c o s} \phi-d )} {\left[ r^{3}+d^{3}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \phi-\left( \frac{a} {d} \right)^{4} d \right)} {\left[ a^{4}+\left( \frac{r} {a} \right)^{2} d^{3}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}
$$
E. $$
\frac{m ( r \operatorname{c o s} \theta-\frac{a} {d} )} {\left[ r^{2}+d^{4}-2 r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \theta-\left( \frac{a} {d} \right)^{2} \right)} {\left[ a^{3}+\left( \frac{r} {a} \right)^{2} d^{4}-2 r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}
$$
F. $$
\frac{m ( r \operatorname{c o s} \theta-d a )} {\left[ r^{2}+d^{2}-2 r d a \operatorname{c o s} \theta\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \theta-\left( \frac{a} {d} \right)^{2} \right)} {\left[ a^{2}+\left( \frac{r} {a} \right)^{2} d-2 r a d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}
$$
G. $$
\frac{m ( r \operatorname{s i n} \theta-d )} {\left[ r^{2}+d^{2}-2 r d \operatorname{s i n} \theta\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{s i n} \theta-\left( \frac{a} {d} \right) d \right)} {\left[ a^{2}+\left( \frac{r} {a} \right)^{3} d^{2}-2 r d \operatorname{s i n} \theta\right]^{\frac{3} {2}}}
$$
H. $$
\frac{m ( \operatorname{c o s} \theta-d r )} {\left[ d^{2}+r^{4}-3 r a \operatorname{c o s} \theta\right]^{\frac{3} {2}}}-\frac{m \left( \operatorname{c o s} \theta-\left( \frac{a} {r} \right)^{2} d \right)} {\left[ a^{4}+\left( \frac{r} {a} \right)^{2} d^{3}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}
$$
I. $$
\frac{m ( r \operatorname{c o s} \phi-d a )} {\left[ r^{4}+d^{4}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}-\frac{m \left( r \operatorname{c o s} \phi-\left( \frac{a} {r} \right)^{2} d r \right)} {\left[ a^{3}+\left( \frac{r} {a} \right)^{3} d^{4}-r d \operatorname{s i n} \theta\right]^{\frac{3} {2}}}
$$
J. $$
\frac{m ( d \operatorname{c o s} \theta-r )} {\left[ r^{3}+d^{3}-2 r d \operatorname{c o s} \phi\right]^{\frac{3} {2}}}-\frac{r \left( d \operatorname{c o s} r-\left( \frac{a} {d} \right) r d \right)} {\left[ a^{4}+\left( \frac{r} {a} \right)^{3} d^{4}-2 r d \operatorname{c o s} \theta\right]^{\frac{3} {2}}}
$$
Answer: C