Electric Field Lines Simulator
Place point charges and watch field lines and equipotentials form in real time � Coulomb's law made visible.
Place Charges
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Presets
Controls: Click canvas ? place charge | Drag ? move charge | Right-click ? remove charge | C clear all | E equipotentials | L field lines
Understanding Electric Fields
Coulomb's Law
The force between two point charges: F = kq1q2/r� where k = 8.99�10? N�m�/C�. Force is repulsive for like charges (both positive or both negative) and attractive for opposite charges. Force doubles when charge doubles; quadruples when separation halves.
Electric Field Strength
The electric field E at any point is the force per unit positive charge: E = F/q = kQ/r�. It is a vector � pointing away from positive charges, toward negative charges. The field tells you the force a charge would experience without needing to place a real charge there.
Superposition Principle
When multiple charges are present, the total field is the vector sum of individual contributions: E_total = Skq?r?^/r?�. Similarly, electric potential is a scalar sum: V_total = Skq?/r?. This is why dipole fields look so different from single-charge fields.
Electric Potential
Potential V is the work done per unit charge to bring a positive test charge from infinity: V = kQ/r. Equipotential surfaces connect points of equal V. Moving along an equipotential requires no work. Field lines always point from high to low potential � "downhill" in the potential landscape.
Key Equations
| Quantity | Symbol | Formula | Units |
|---|---|---|---|
| Coulomb force | F | kq1q2 / r� | N |
| Electric field | E | F/q = kQ/r� | N/C = V/m |
| Electric potential | V | kQ/r = W/q | V (volt) |
| Electric PE | U | kq1q2/r = qV | J |
| Coulomb constant | k | 1/(4pe0) � 8.99�10? | N�m�/C� |
| Permittivity of free space | e0 | 8.85�10?�� C�/(N�m�) | F/m |
| E from potential | E | -?V = -dV/dr | V/m |
| Field inside conductor | E | 0 | N/C |
Field Line Rules
Direction
Field lines start on positive charges and end on negative charges (or go to infinity if no negative charge is present). A positive test charge would follow the field line direction.
Density
The density of field lines in a region indicates field strength. Closely-spaced lines = strong field. Widely-spaced lines = weak field. Near a point charge, lines are densest close to the charge.
Never Cross
Field lines can never cross each other. If they did, the field would have two directions at that point � physically impossible. At saddle points between equal charges, field lines approach but never cross.
Perpendicular to Equipotentials
Field lines are always exactly perpendicular to equipotential surfaces. This is because E = -?V � the field points in the direction of steepest potential drop, which is always perpendicular to constant-potential surfaces.
Common Configurations
Electric Dipole
Equal and opposite charges separated by distance d. The dipole moment p = qd. Field lines arch from + to -, forming characteristic cardioid-like loops. At large distances, dipole field falls as 1/r� (faster than 1/r� for a single charge).
Parallel Plate Capacitor
Two rows of opposite charges approximate a parallel plate capacitor. Between the plates, the field is nearly uniform and perpendicular to the plates. Field strength E = s/e0 = V/d in the ideal infinite-plate limit.
Quadrupole
Four charges (+ - + -) arranged at corners of a square. More complex field topology with multiple neutral points. Quadrupole fields fall as 1/r4 at large distances. Used in particle accelerator focusing magnets.
Gauss's Law
The total electric flux through any closed surface equals the enclosed charge divided by e0: ?E�dA = Q_enc/e0. Powerful for symmetric charge distributions � gives E immediately for spheres, cylinders, and planes.
Worked Example
Step 1: Single Charge
A +5 nC charge sits at the origin. Find E and V at r = 0.10 m.
E = kQ/r� = (8.99�10? � 5×10⁻⁹) / 0.01 = 4495 N/C
V = kQ/r = (8.99�10? � 5×10⁻⁹) / 0.10 = 449.5 V
Step 2: Second Charge
A -5 nC charge is placed at (0.20 m, 0). Find E at the midpoint (0.10 m, 0).
From +5 nC: E1 = 4495 N/C pointing right (+x).
From -5 nC: E2 = 4495 N/C pointing right (+x toward - charge).
E_total = 8990 N/C in +x direction.
Step 3: Force on Test Charge
A +1 nC test charge is placed at the midpoint. What force does it experience?
F = qE = 1�10⚡ � 8990 = 8.99�10?6 N (toward -5 nC charge).
Step 4: Potential Energy
V at midpoint: V1 = 449.5 V, V2 = kQ/r = 8.99�10?�(-5×10⁻⁹)/0.10 = -449.5 V.
V_total = 449.5 - 449.5 = 0 V (midplane of dipole is at V=0).
PE of test charge: U = qV = 1�10📚 � 0 = 0 J.
Curriculum Links
| Level | Topic | Key Concepts Covered |
|---|---|---|
| GCSE Physics | Electrostatics | Static charge, attraction/repulsion, field lines direction, spark discharge |
| A-Level Physics | Electric Fields | Coulomb's law, E = kQ/r�, V = kQ/r, equipotentials, E = -?V/?r, capacitors |
| IB Physics HL | Fields (Topic 10) | Coulomb force, field strength, potential, work done in field, capacitance |
| AP Physics C | Electrostatics | Gauss's law, field from superposition, potential energy, conductors in E fields |
| University Year 1 | Electromagnetism | Maxwell's equations (static), multipole expansion, image charges, conductors |
Frequently Asked Questions
Why do electric field lines never cross?
If two field lines crossed at a point, the electric field would simultaneously point in two different directions at that point � which is physically impossible. The field at any point is uniquely determined by the superposition of all charges. Crossing would imply the field is multivalued, violating the uniqueness of the Coulomb force.
What happens to field lines inside a conductor?
In electrostatic equilibrium, the electric field inside a conductor is exactly zero. Free electrons rearrange themselves on the surface until the internal field cancels out. All field lines terminate (or originate) on the surface charges. This is why a Faraday cage shields the interior from external fields � and why car bodies protect people from lightning.
How is electric potential energy different from electric potential?
Electric potential V = kQ/r is a property of the field at a point, with units of volts (J/C). Electric potential energy U = qV is the energy stored when a specific charge q is placed at that point. Potential is a property of space; potential energy involves placing a charge in that space. Analogy: gravitational field (g) vs. gravitational PE (mgh).
Why are equioptential lines perpendicular to field lines?
The electric field is mathematically defined as E = -?V (the negative gradient of potential). The gradient of a scalar always points in the direction of maximum increase. Equipotential surfaces are surfaces of constant V, so moving along them changes V by zero � meaning you move perpendicular to the gradient. Therefore E is always perpendicular to equipotentials.
What is the significance of the inverse-square law?
The 1/r� dependence of Coulomb's law (and Newton's gravity) arises from geometry: in 3D space, electric flux spreads over a spherical surface whose area grows as 4pr�. Since total flux is conserved (Gauss's law), field strength must decrease as 1/r�. This 1/r� form is not arbitrary � it is a direct consequence of living in three spatial dimensions.
Related Simulations
Further Reading
- Electromagnetic Waves and Maxwell's Equations � how static fields extend to dynamic EM waves
- Special Relativity and Spacetime � how electric and magnetic fields transform between frames
- Thermodynamics and Heat Engines � thermal analogy with potential fields
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