Constant electric field in vacuum

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• Strength and potential of the field of a point charge $q$:$$ \vec{E}=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r^3}\vec{r},   \phi=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r}.$$
• Relation between field strength and potential:$$\vec{E}=- \vec{\nabla}\phi. $$ i.e. field strength is equal to the anti gradient of the potential.
• Gauss's theorem and circulation of the vector $\vec{E}$:$$\oint \vec{E}.d\vec{S}=\frac{q}{{\varepsilon}_0}, \oint  \vec{E}. d\vec{r}=0.$$
• Potential and strength of the field of a point dipole with electric moment $\vec{p}$:$$\phi =\frac{1}{4\pi {\varepsilon}_0}\frac{\vec{p}.\vec{r}}{r^3}, E=\frac{1}{4\pi {\varepsilon}_0}\frac{p}{r^3}\sqrt{1+3{\cos}^2 \theta}.$$ where $\theta$ is the angle between the vectors $\vec{r}$ and $\vec{p}$.
• Energy $W$ of the dipole $\vec{p}$ in an external electric field, and the moment $\vec{N}$ of forces acting on the dipole:$$W =-\vec{p}.\vec{E}, \vec{N}=\vec{p}\times \vec{E}.$$
• Force $\vec{F}$ acting on a dipole, and its projection $F_x$:$$\vec{F}=p\frac{\partial \vec{E}}{\partial l},F_x=\vec{p}.\vec{\nabla} E_x $$ where $\partial \vec{E}/\partial l$ is the derivative of the vector $\vec{E}$ with respect to the dipole direction, $\nabla E_x$ is the gradient of the function $E_x$.