Electricity is often described by the electric field. The electric field is an invisible field that exists around every charge. The electric field is a vector field, where every point in the field has a magnitude and a direction. The total electric field at any point is the vector sum of the electric fields due to all charges that are present in the system.
The electric field at a point \( (x,y,x) \) is defined by the following:
$$ \vec{E} (x,y,z) = \frac{\vec{F}}{q_0}, $$
where \( \vec{F} \) is the force that acts on a test charge \( q_0 \) placed at the point \( (x,y,z) \). The electric field points in the direction of the force on a positive test charge.
We can visualize electric fields by means of electric field lines. We imagine that these lines come out of positive charges, and they end on the negative charges. At a given point, the electric field direction is tangent to the eletric field line passing through that point, and the magnitude of the electric field at a point is propoeration to the density of lines.
The electric field of a point charge \( q \) at a point \( (x,y,z) \), and a distance \( r \) from the charge is:
$$ \vec{E} = k \frac{q}{r^2} \hat{r} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r} $$
where \( \hat{r} \) is a unit vector directed radially away from the charge and directed along the electric field lines.