(lines with arrows) of a nearby positive charge
(+) causes the mobile charges in conductive objects to separate due to
electrostatic induction. Negative charges
(blue) are attracted and move to the surface of the object facing the external charge. Positive charges
(red) are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore, the electrostatic field everywhere inside a conductive object is zero, and the
electrostatic potential is constant. The electric field, \mathbf E, in units of
newtons per
coulomb or
volts per meter, is a
vector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity). It is defined as the electrostatic force \mathbf F on a hypothetical small
test charge at the point due to Coulomb's law, divided by the charge q : \mathbf E = {\mathbf F \over q}
Electric field lines are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point. A collection of n particles of charge q_i, located at points \mathbf r_i (called
source points) generates the electric field at \mathbf r (called the
field point) of: states that "the total
electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total
electric charge enclosed by the surface." Many numerical problems can be solved by considering a
Gaussian surface around a body. Mathematically, Gauss's law takes the form of an integral equation: : \Phi_E = \oint_S\mathbf E\cdot \mathrm{d}\mathbf A = {Q_\text{enclosed}\over\varepsilon_0} = \int_V{\rho\over\varepsilon_0}\mathrm{d}^3 r, where \mathrm{d}^3 r =\mathrm{d}x \ \mathrm{d}y \ \mathrm{d}z is a volume element. If the charge is distributed over a surface or along a line, replace \rho\,\mathrm{d}^3r by \sigma \, \mathrm{d}A or \lambda \, \mathrm{d}\ell. The
divergence theorem allows Gauss's law to be written in differential form: : \nabla\cdot\mathbf E = {\rho\over\varepsilon_0}. where \nabla \cdot is the
divergence operator.
Poisson and Laplace equations The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density
ρ: : {\nabla}^2 \phi = - {\rho\over\varepsilon_0}. This relationship is a form of
Poisson's equation. In the absence of unpaired electric charge, the equation becomes
Laplace's equation: : {\nabla}^2 \phi = 0, == Electrostatic approximation ==