Physics · Electromagnetism · Electrostatics
Electric Field Calculator
Calculates the electric field strength produced by a point charge at a given distance using Coulomb's law.
Calculator
Formula
E is the electric field strength in newtons per coulomb (N/C or V/m); k is Coulomb's constant (8.9875 × 10⁹ N·m²/C²); q is the source charge in coulombs (C); r is the distance from the charge to the field point in metres (m); ε₀ is the permittivity of free space (8.854 × 10⁻¹² C²/N·m²).
Source: Griffiths, D.J. — Introduction to Electrodynamics, 4th ed., Eq. 2.4. NIST CODATA 2018 for physical constants.
How it works
An electric field is a region of space around a charged object where another charge experiences a force. The concept was formalized by Michael Faraday and mathematically expressed through Coulomb's law. The electric field E at a point in space represents the force per unit positive test charge placed at that point, making it a property of the source charge and geometry alone — independent of any test charge placed there.
The governing formula is E = kq / r², where k = 8.9875 × 10⁹ N·m²/C² is Coulomb's constant (equal to 1 / (4πε₀)), q is the source charge in coulombs, and r is the distance from the source charge to the field point in metres. The field is a vector quantity: it points radially outward from positive charges and radially inward toward negative charges. The magnitude given by this formula applies in a vacuum or air (with negligible dielectric effect). The electric potential V = kq / r is also computed, representing the potential energy per unit charge at that location.
Practical applications include designing capacitors, modelling electrostatic precipitators, calculating fields near high-voltage power lines, analysing electron beam optics in cathode ray tubes, and understanding the behaviour of ions in mass spectrometers. In semiconductor physics, electric field calculations underpin the analysis of p-n junction depletion regions and transistor gate fields. The field strength output in N/C is numerically equivalent to V/m, connecting directly to voltage gradient measurements in engineering contexts.
Worked example
Suppose a proton carries a charge of q = +1.602 × 10⁻¹⁹ C (the elementary charge), and you want to find the electric field at a distance of r = 5.29 × 10⁻¹¹ m (the Bohr radius, roughly the distance from the proton to the electron in a hydrogen atom).
Step 1 — Identify constants and values:
k = 8.9875 × 10⁹ N·m²/C²
q = 1.602 × 10⁻¹⁹ C
r = 5.29 × 10⁻¹¹ m
Step 2 — Apply the formula:
E = kq / r² = (8.9875 × 10⁹ × 1.602 × 10⁻¹⁹) / (5.29 × 10⁻¹¹)²
Step 3 — Calculate numerator:
8.9875 × 10⁹ × 1.602 × 10⁻¹⁹ = 1.4397 × 10⁻⁹ N·m²/C
Step 4 — Calculate denominator:
(5.29 × 10⁻¹¹)² = 2.798 × 10⁻²¹ m²
Step 5 — Divide:
E = 1.4397 × 10⁻⁹ / 2.798 × 10⁻²¹ ≈ 5.14 × 10¹¹ N/C
This enormous field strength is consistent with atomic-scale electrostatics and explains the strong binding between the proton and electron in a hydrogen atom. The electric potential at this distance is V = kq / r ≈ 27.2 V, corresponding to twice the hydrogen ionization energy (13.6 eV) — a well-known result in quantum chemistry.
Limitations & notes
This calculator assumes the source is an ideal point charge in a vacuum (or air, where the dielectric constant ε ≈ 1). For charges embedded in a dielectric medium, the result must be divided by the relative permittivity εᵣ of the medium (e.g., εᵣ ≈ 80 for water). The formula breaks down in the immediate vicinity of a real charge distribution (r → 0), as real charged objects have finite size. For multiple charges, superposition must be applied — the total field is the vector sum of individual fields, which this calculator does not handle. Extended charge distributions (line charges, surface charges, volume charges) require integration using Gauss's law rather than the point-charge formula. At very high field strengths (above ~10¹⁸ V/m), quantum electrodynamic (Schwinger) effects become relevant and classical Coulomb's law fails. This tool computes field magnitude only; the direction is radially outward for positive q and radially inward for negative q.
Frequently asked questions
What is the difference between electric field and electric potential?
The electric field E is a vector quantity representing force per unit charge (N/C or V/m), describing how a test charge would accelerate at a given point. Electric potential V is a scalar quantity representing potential energy per unit charge (in volts), related to E by E = -dV/dr for a point charge. While the field tells you the force direction and magnitude, the potential is often easier to sum for multiple charges because scalars add algebraically.
Why does the electric field decrease with the square of the distance?
The inverse-square relationship arises because electric field lines spread out over the surface area of a sphere (4πr²) surrounding the source charge. As distance doubles, the same number of field lines is spread over four times the area, so the field intensity decreases by a factor of four. This is the same geometric reason that gravitational and light intensity fields also follow inverse-square laws.
How do I calculate the electric field from a negative charge?
Simply enter a negative value for q. The formula E = kq/r² will return a negative result, indicating the field points toward the source charge rather than away from it. By convention, the magnitude |E| gives the field strength, and the sign (or vector direction) tells you the orientation — inward for negative charges and outward for positive charges.
Can this calculator be used for charges in water or other materials?
The calculator as presented assumes free space (vacuum). To account for a dielectric medium, divide the computed electric field by the relative permittivity εᵣ of the material. For water, εᵣ ≈ 80, so the field is about 80 times weaker than in vacuum. For glass, εᵣ ≈ 4–10 depending on the type. This modification also applies to the electric potential output.
What units should I use for charge when entering microcoulombs or nanocoulombs?
This calculator uses SI base units, so charge must be entered in coulombs (C). Convert before entering: 1 microcoulomb (μC) = 1 × 10⁻⁶ C, 1 nanocoulomb (nC) = 1 × 10⁻⁹ C, and 1 picocoulomb (pC) = 1 × 10⁻¹² C. Use scientific notation in the input field, such as 1e-6 for 1 μC or 2.5e-9 for 2.5 nC.
Last updated: 2025-01-15 · Formula verified against primary sources.