Electric Field Calculator | Point Charge & Plates
Calculate electric field strength from point charges (E = kQ/r²), parallel plates (E = V/d), and electrostatic force (F = qE) with unit conversion and air-breakdown indicator.
All calculations use NIST-defined constants: k = 8.9876 × 10⁹ N·m²/C², ε₀ = 8.854 × 10⁻¹² F/m. Computations run live in your browser, no data is sent to any server.
What Is the Electric Field Calculator | Point Charge & Plates?
The electric field E at a point in space is defined as the electrostatic force per unit positive test charge placed at that point: E = F/q. It is a vector, it has both magnitude and direction, pointing away from positive source charges and toward negative ones. The unit N/C is identical to V/m, connecting field strength directly to potential gradients.
For a single point charge Q, Coulomb's law gives E = kQ/r², where k = 8.9876 × 10⁹ N·m²·C⁻² is derived from the NIST 2018 CODATA recommended value of the permittivity of free space (ε₀ = 8.8542 × 10⁻¹² F·m⁻¹). The field falls off as the square of distance, so doubling r reduces E by a factor of four. The accompanying electric potential is V = kQ/r, which falls off only linearly with distance.
Between two large parallel plates with a voltage V across gap d, the field is approximately uniform: E = V/d. This geometry is used in capacitors, electron guns, and particle accelerators. The energy stored per unit volume of the field is u = ε₀E²/2, directly useful for capacitor energy calculations.
When multiple charges are present, the superposition principle applies: the total field at any point is the vector sum of contributions from every individual charge. This principle underlies all charge distribution calculations, line charges, surface charges, and volume distributions alike.
Formula
| Symbol | Name | Description |
|---|---|---|
| E | Electric field strength | Force per unit positive charge; N/C = V/m |
| k | Coulomb's constant | 8.9876 × 10⁹ N·m²·C⁻²; k = 1/(4πε₀) |
| Q | Source charge | Charge creating the field; coulombs [C] |
| r | Distance from charge | Radial distance from the point charge to field point; metres [m] |
| ε₀ | Permittivity of free space | 8.8542 × 10⁻¹² F·m⁻¹; from NIST CODATA 2018 |
| V | Potential difference | Voltage across parallel plates; volts [V] |
| d | Plate separation | Gap between plates in uniform-field geometry; metres [m] |
| F | Electrostatic force | Force on a charge q in field E; newtons [N] |
| q | Test / probe charge | Charge experiencing the force; coulombs [C] |
| σ | Surface charge density | ε₀E; charge per unit area on plates; C/m² |
| u | Electric energy density | ε₀E²/2; energy stored per unit volume; J/m³ |
How to Use
- 1Select calculation mode: Choose "Point Charge" (Coulomb's law), "Parallel Plates" (uniform field), or "Force on Charge" (F = qE) using the mode tabs at the top.
- 2Choose what to solve for: Each mode offers three solve-for options via button group, select the unknown quantity. The locked field (shown dimmed) will be calculated for you.
- 3Enter known values: Type values into the active input boxes. For charge Q and distance r, choose appropriate units (pC, nC, μC, C and nm, μm, mm, cm, m) from the dropdowns.
- 4Load a preset (optional): Click any preset button (e.g., "1 μC at 10 cm" or "Proton at 1 Å") to auto-fill representative values and see a worked result immediately.
- 5Press Calculate or Enter: Click "Calculate" or press Enter while in any input field. Results appear in SI units with scientific notation for very large or small values.
- 6Read primary results: Point charge mode shows E, the electric potential V, and forces on an electron and proton. Plates mode shows E, surface charge density σ, energy density u, and an air-breakdown indicator.
- 7Check the step trace: Expand the calculation steps section to see every intermediate result: unit conversions, formula substitution, and final answer, useful for verifying hand calculations.
- 8Reset or recall: Press Reset or Esc to clear inputs. Your last valid inputs are automatically saved to localStorage and restored on the next visit.
Example Calculation
Example 1: Electric field from a point charge
A charge Q = 2 μC sits at the origin. Find the electric field at r = 0.5 m and the force on an electron at that point.
Example 2: Parallel plate capacitor field
A capacitor has plates separated by d = 1 mm with a potential difference of V = 120 V. Find E, σ, and energy density.
Example 3: Solving for distance (inverse problem)
At what distance from Q = 10 nC does the field equal E = 1000 N/C?
Understanding Electric Field | Point Charge & Plates
What Is an Electric Field?
An electric field is a region of space in which an electrically charged particle experiences a force. Introduced by Michael Faraday as a way to describe action at a distance, the field concept replaced the idea of charges acting directly on one another across empty space. The field exists whether or not a test charge is present to detect it, it is a physical property of the space surrounding any charge distribution.
Electric Field Lines
- ›Field lines originate on positive charges and terminate on negative charges.
- ›The density of lines at any point is proportional to the field strength, lines crowd together where E is large.
- ›Field lines never cross (the field has a unique direction at every point).
- ›For a point charge, lines radiate uniformly in all directions; for parallel plates, they run parallel and uniform between the plates.
- ›A conductor in electrostatic equilibrium has no internal field; excess charge resides on the surface and the field just outside is perpendicular to the surface.
Gauss's Law
Gauss's law states that the total electric flux through any closed surface equals the enclosed charge divided by ε₀: Φ = Q_enc / ε₀. It is mathematically equivalent to Coulomb's law for static charges, but is far more powerful when dealing with symmetric charge distributions. A spherical Gaussian surface around a point charge immediately recovers E = kQ/r²; a cylindrical surface around an infinite line charge gives E = λ/(2πε₀r); a pillbox surface at a plane gives E = σ/(2ε₀). These results underlie much of classical electrostatics.
Physical Constants Used
This calculator uses NIST CODATA 2018 recommended values for all fundamental constants: k = 8.987 551 7923 × 10⁹ N·m²·C⁻², ε₀ = 8.854 187 817 × 10⁻¹² F·m⁻¹, and the elementary charge e = 1.602 176 634 × 10⁻¹⁹ C (exact under the 2019 SI redefinition). Values are sourced from the NIST Physical Reference Data database at physics.nist.gov.
Practical Applications
- ›Capacitors & energy storage: Parallel plate field E = V/d determines capacitance C = ε₀A/d and stored energy U = ½CV².
- ›Cathode ray tubes & electron guns: Uniform fields accelerate electrons to precise velocities; E controls deflection.
- ›Particle accelerators: Alternating electric fields in linacs and cyclotrons impart energy to charged particles each pass.
- ›Electrostatic precipitators: High fields impart charge to particles in flue gas so they migrate to collection plates.
- ›Inkjet printers: Charged droplets are steered by deflecting fields of a few kV/cm to hit the correct pixel position.
- ›Lightning rods & corona discharge: Sharp conductors concentrate field lines; when E exceeds ≈ 3 MV/m, air ionises and conducts.
Air Electrical Breakdown
Air begins to ionise when the electric field exceeds approximately 3 MV/m (3 × 10⁶ V/m)under standard conditions. This is called the dielectric strength of air. Once ionised, air becomes conductive and a spark or lightning discharge occurs. The parallel plates mode shows a breakdown indicator: below 80% of this threshold is safe, 80–100% is a caution zone, and above 3 MV/m the display warns of likely discharge. Humidity, pressure, and gap geometry all shift this threshold in real systems.
Frequently Asked Questions
What are the units of electric field strength?
Electric field has two equivalent SI units: N/C (newtons per coulomb) and V/m (volts per metre). They are identical, the choice is a matter of context.
- • N/C is natural when using F = qE to find force on a charge.
- • V/m is natural when using E = V/d for parallel plates or when connecting to potential gradients.
- • Both units reduce to kg·m·s⁻³·A⁻¹ in base SI, confirming they are the same.
What is Coulomb's constant k and where does it come from?
Coulomb's constant is defined as k = 1 / (4πε₀) where ε₀ is the permittivity of free space.
- • NIST 2018 value: k = 8.987 551 7923 × 10⁹ N·m²·C⁻²
- • ε₀: 8.854 187 817 × 10⁻¹² F·m⁻¹ (also called the electric constant)
- • Since the 2019 SI redefinition, ε₀ is no longer exact but is derived from c and μ₀. Its value is known to 10 significant figures.
This calculator uses the full NIST CODATA 2018 precision, not the rounded textbook value of 9 × 10⁹.
What is the difference between electric field and electric potential?
They describe different aspects of the same electromagnetic reality:
- • Electric field E is a vector, it has magnitude and direction at every point. It equals the force per unit positive charge.
- • Electric potential V is a scalar, it is the potential energy per unit charge (joules per coulomb = volts).
- • They are related by E = −∇V: the field points from high to low potential, and its magnitude equals the rate of change of V with distance.
- • For a point charge: E = kQ/r² but V = kQ/r, potential falls off more slowly than field strength.
Why does the electric field drop as 1/r² for a point charge?
The 1/r² dependence is a direct consequence of geometry in three-dimensional space, not a special property of electricity.
- • A point charge radiates field lines equally in all directions. These lines pass through a sphere of radius r centred on the charge.
- • As r doubles, the sphere's surface area quadruples (A = 4πr²), so the same number of field lines are spread over four times the area.
- • Field strength is proportional to line density: E ∝ 1/r²
- • Gauss's law makes this rigorous: the total flux Φ = Q/ε₀ is constant, so E = Q/(4πε₀r²) = kQ/r².
Gravity obeys the same inverse-square law for exactly the same geometric reason.
How do I calculate the field between parallel plates?
For two large conducting plates with a voltage difference V and separation d:
- The field is uniform between the plates (edge effects neglected): E = V / d
- Measure V in volts and d in metres; E is in V/m.
- Surface charge density on each plate: σ = ε₀E in C/m²
- Energy stored per unit volume of field: u = ε₀E² / 2 in J/m³
- Total stored energy in a capacitor of area A and gap d: U = ½ε₀E²·Ad = ½CV²
The field is directed from the positive plate to the negative plate.
What is air electrical breakdown and why does it happen?
Air has a dielectric strength of approximately 3 MV/m (3 × 10⁶ V/m) under standard conditions (1 atm, room temperature, dry).
- • When E exceeds this threshold, free electrons in the air are accelerated fast enough to ionise neutral molecules on collision.
- • This triggers an avalanche (Townsend discharge), making air temporarily conductive, visible as a spark or lightning.
- • Factors that lower breakdown voltage: high humidity, reduced pressure, sharp electrode geometry (which concentrates field lines).
- • Practical rule: a 1 mm gap breaks down at roughly 3 kV; a 1 cm gap at roughly 30 kV.
The plates mode in this calculator shows a real-time breakdown percentage and colour-coded warning as you approach this threshold.
Can the electric field be zero even when charges are present?
Yes, the superposition principle allows fields from multiple charges to cancel at specific points.
- • Two equal positive charges +Q separated by distance 2d create a zero field exactly midway between them (fields point in opposite directions and cancel).
- • An electric dipole (+Q and −Q) has a zero-field point on the perpendicular bisector plane only at infinity; along the axis, the field is never zero between the charges.
- • Inside a conductor in electrostatic equilibrium, free charges redistribute until the internal field is exactly zero, any net charge resides on the surface.
Finding the null points of multi-charge systems requires solving the vector sum of individual fields, which this calculator supports for single charges and uniform fields.
How are electric field calculations used in real engineering?
Electric field calculations appear across many engineering disciplines:
- • Capacitor design: E = V/d sets the required plate spacing to avoid breakdown at operating voltage.
- • High-voltage transmission: Corona discharge losses on power lines are minimised by controlling the surface field below 1–2 MV/m.
- • MRI & particle physics: Uniform fields accelerate, deflect, or focus charged particle beams with precise energy control.
- • Semiconductor devices: Gate fields in MOSFETs control channel conductivity; oxide breakdown limits dictate maximum gate voltage.
- • Electrospray & mass spectrometry: Strong fields at needle tips ionise analyte molecules for mass analysis.