Constant electric field in vacuum

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

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