# Find the equipotential surface

Electricity and Magnetism Level pending

A small electric dipole with a dipole moment $$p= 1.2\times 10^{-8}~\mbox{m}\cdot\mbox{C}$$ is placed in an external electric field $$E_{0}=1~\mbox{V/m}$$ which is parallel to the dipole, that is, $\vec{p} \parallel \vec{E_{0}}.$ It turns that in this case one of the equipotential surfaces enclosing the dipole is a sphere. Find the radius $$R$$ in meters of this sphere. The electric potential created by a point dipole at a point P is given by the formula $\phi(\vec{r})=\frac{1}{4\pi \varepsilon_{0}} \frac{\vec{p}\cdot \vec{r}}{r^{3}}$ where $$\vec{r}$$ is the radius vector from the center of the dipole to the point P.

Details and assumptions

$k=\frac{1}{4\pi \epsilon_{0}}= 9\times 10^{9}~\text{m/F}$

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