The electric field is the trick that lets us stop talking about charges pulling on charges directly, and instead say that a charge fills the space around it with a condition — a field — that any other charge then simply responds to. Bring a tiny positive test charge q₀ to a point and measure the force F it feels; the field there is
E = F/q₀ — the force per unit positive charge, in N·C⁻¹.
Dividing out q₀ is the whole point: E describes the space, not the visitor. Once you know E at a point, the force on any charge q placed there is just F = qE — along E if q is positive, opposite if negative.
Field of a point charge. Put a charge Q at the origin. The field it makes a distance r away has magnitude E = kQ/r² with k = 9 × 10⁹, pointing radially outward from a positive Q and inward toward a negative one. It is Coulomb's law with one q₀ stripped off — same inverse-square falloff.
Superposition. Fields add as vectors. The net field of many charges is the vector sum E = E₁ + E₂ + …, each term computed as if the others were absent. Resolve into components; never add magnitudes blindly.
Field lines picture this: they start on positive charge, end on negative, never cross (the field has one direction at each point), and crowd together where the field is strong. The tangent gives the field's direction.
The electric dipole is two equal and opposite charges +q and −q a small distance d apart. Its strength is the dipole moment p = qd, a vector pointing from −q to +q. Because the two charges nearly cancel, their fields don't fall as 1/r² — the leading terms subtract, leaving a faster 1/r³ decay. On the axis (the line through both charges) the field is E = 2kp/r³; on the equatorial line (perpendicular bisector) it is E = kp/r³, pointing opposite to p. Hence the memorable 2:1 axial-to-equatorial ratio at equal distance — the axial point "sees" both charges reinforcing, the equatorial point sees them partly cancel.
A dipole in a uniform field feels no net force (the two charges pull equally and oppositely), but it feels a torque τ = pE sinθ, or τ = p × E, that twists it toward alignment with E. Its orientation energy is U = −pE cosθ = −p·E: lowest (−pE) when aligned, highest (+pE) when anti-aligned — the physics behind why dipoles rotate to line up.
Motion of a charge. A charge released in a field accelerates by Newton's law: a = qE/m. In a uniform field this is constant, so the charge moves like a projectile — straight-line speed-up along E, or a parabola if it enters sideways. Read every problem as one of these ideas: field of a source, superposition, or a charge's response.