Electric Fields & Field Lines

The field of a point charge

A charge fills the space around it with an electric field — the force per unit charge that a small positive test charge q₀ would feel, F = q₀E. Its strength falls off as one over distance squared (Coulomb's inverse-square law), pointing radially away from a positive charge and towards a negative one. Drag the +q probe on the stage and the force arrow you see is exactly this: q times the local field. The constant 1/4πε₀ ≈ 8.99×10⁹ N·m²·C⁻² sets the scale in SI units.

Superposition — the whole game

With several charges the fields simply add as vectors — the principle of superposition, and it is exact because Maxwell's equations are linear. Every arrow, every field line on the stage is that vector sum evaluated at that point; drag a charge and the entire pattern re-solves in real time. There is never a new rule to learn — every configuration in physics, however intricate, is just this sum of inverse-square terms.

Potential and field: E = −∇V

The potential V is the work done per unit charge to bring a test charge in from infinity — a single number at each point, far easier to add than vectors because it has no direction. The field is its steepest downhill slope: E points from high potential to low, and its size is how fast V changes with distance. Surfaces of constant V (equipotentials) are therefore always perpendicular to the field lines, and no work is done moving a charge along one.

Gauss's law — why inverse-square

The number of field lines piercing any closed surface depends only on the charge inside it — the flux is Q/ε₀ and nothing else. Because a point charge sheds its lines evenly over the 4πr² of an expanding sphere, the field must thin out as 1/r²: the inverse-square law is geometry, the surface area of space. Gauss's law is the deep reason the lines never simply stop in empty space — they can only begin on a positive charge and end on a negative one.

Reading the lines, reading the patterns

Field lines are tangent to the field everywhere, so they show its direction (the little arrowheads); where they crowd together the field is strong, where they fan out it is weak, and the count leaving a charge is proportional to its size. Two like charges push their lines apart and leave a neutral point between them where the fields exactly cancel; a dipole gives the looping +→− pattern; a quadrupole shows four lobes. Lines can never cross, because the field has one definite direction at every point.

Numerically verified

• Field of a single charge follows the inverse-square law |E| = k|q|/r² to machine precision (away from the softening radius)
• Potential of a single charge follows V = kq/r; equipotential structure is the gradient of what's plotted
• Superposition is exact — the rendered field is the vector sum of every charge's contribution, recomputed each frame
• Field lines integrate the field direction and terminate on opposite charges, with line count ∝ |q|

Scientific references

1. [1]D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press (2017)
2. [2]E. M. Purcell & D. J. Morin, Electricity and Magnetism, 3rd ed., Cambridge University Press (2013)
3. [3]NCERT Physics Class XII, Part 1, Ch. 1–2: Electric Charges, Fields and Potential