Electric Field Calculator

Calculate electric field strength, force, source charge, or distance using E = kQ/r² and F = qE.

⚡ Electricity📐 E = kQ/r²🔌 Electric Field
Source charge (Q) Coulombs
Distance (r) m
⚠️ Enter valid numbers. Q must be non-zero.

What Is Electric Field?

The electric field (E) at a point in space quantifies the electric force per unit positive charge that would be experienced at that point: E = F/q. It is a vector quantity pointing away from positive charges and toward negative charges. The SI unit is N/C (newtons per coulomb), equivalent to V/m (volts per meter).

For a point charge Q, the electric field at distance r is given by Coulomb's law: E = kQ/r², where k = 8.99×10⁹ N·m²/C² is Coulomb's constant (= 1/(4πε₀) where ε₀ = 8.85×10⁻¹² F/m). Field strength falls off as 1/r² — doubling distance reduces field to 1/4 of its original strength.

The force on any charge q placed in an electric field E is simply F = qE. Positive charges are pushed in the direction of E; negative charges are pushed opposite to E. Electric fields exert forces without contact — this 'action at a distance' was mysterious to Newton but is fully explained by Maxwell's equations.

Electric field lines visualize the field: they point from positive to negative charges, are denser where the field is stronger, and never cross. The relationship between field and voltage is E = −dV/dr — field points from high to low potential. For a uniform field between parallel plates: E = V/d.

Formula Reference Table

Solve ForFormulaNotes
E-field (point charge)E = kQ/r²k = 8.99×10⁹ N·m²/C²
Force on test chargeF = q · Eq = test charge; F in Newtons
From Coulomb's LawF = kQq/r²Between two point charges
Parallel platesE = V/dV = voltage, d = plate separation
E from potentialE = −dV/drField = −gradient of potential
Electric fluxΦ = E · AGauss's Law: Φ = Q_enclosed/ε₀

3 Worked Examples

Example 1
E-field from a Proton

Find E-field 1 nm from a proton (q = 1.6×10⁻¹⁹ C).

  • k = 8.99×10⁹ N·m²/C²
  • E = kQ/r² = 8.99×10⁹ × 1.6×10⁻¹⁹ / (10⁻⁹)²
  • E = 1.438×10⁻⁹ / 10⁻¹⁸ = 1.44×10⁹ N/C
  • This is ~1.44 GN/C — enormous at atomic distances
✓ E = 1.44×10⁹ N/C at 1 nm from proton
Example 2
Force on Electron in Field

An electron (q = −1.6×10⁻¹⁹ C) is in a 1,000 V/m uniform field.

  • F = qE = (−1.6×10⁻¹⁹) × 1,000
  • F = −1.6×10⁻¹⁶ N (force opposes field direction)
  • Acceleration: a = F/m_e = 1.6×10⁻¹⁶ / 9.11×10⁻³¹ = 1.76×10¹⁴ m/s²
✓ F = 1.6×10⁻¹⁶ N; a = 1.76×10¹⁴ m/s²
Example 3
Parallel Plate Capacitor

10 μC charge, plates 5 cm apart, 1 cm × 1 cm. Find E.

  • Surface charge density σ = Q/A = 10⁻⁵/(0.01)² = 0.1 C/m²
  • E = σ/ε₀ = 0.1/(8.85×10⁻¹²) = 1.13×10¹⁰ N/C
  • Or: if V = 500 V across 5 cm: E = V/d = 500/0.05 = 10,000 V/m
✓ E = 10,000 V/m (for 500 V, 5 cm gap)

Real-World Applications

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Lightning
Lightning forms when charge separation creates E > 3×10⁶ V/m (breakdown strength of air). Positive leaders from clouds and negative from ground meet, and the enormous electric field accelerates electrons to ionize air, creating the luminous channel.
📱
Touchscreens
Capacitive touchscreens use the electric field generated by the display grid. A finger perturbs the local field, changing capacitance. The field-sensing resolution determines touch accuracy.
🔬
Electron Microscopes
TEMs and SEMs use electric fields to accelerate electrons to keV energies. Magnetic fields focus the beam. Electric field E = V/d across gun plates determines electron kinetic energy eV.
⚡
Van de Graaff Generators
A Van de Graaff generator builds static charge until E at the sphere surface exceeds air breakdown (~3 MV/m). Sparks then discharge to ground. Large research generators reach 25 MV.
🌩️
Atmospheric Electricity
Earth's fair-weather electric field is ≈100 V/m pointing downward (negative Earth, positive ionosphere). This creates the global electric circuit — maintained by thunderstorms occurring worldwide.

Common Mistakes to Avoid

⚠️
Forgetting k vs. 1/(4πε₀)

k = 8.99×10⁹ N·m²/C² = 1/(4πε₀). These are identical. Using k = 9×10⁹ is an acceptable approximation; using 9×10⁶ or other wrong values gives large errors.

⚠️
Using r instead of r²

E = kQ/r², not kQ/r. Forgetting the square is the most common error. The inverse-square law means doubling distance reduces E to 1/4, not 1/2.

⚠️
Ignoring charge sign

Field direction depends on Q sign. Positive Q: field points outward. Negative Q: field points inward. When calculating F = qE, the signs of both q (test charge) and E (from source) determine force direction.

⚠️
Confusing E (N/C) and V (volts)

Electric field E = V/m = N/C. They have the same units but measure different things: E is force per unit charge; V (voltage) is energy per unit charge.

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Treating superimposed fields wrong

Multiple charges: E_total = ΣEᵢ (vector sum). You must add E vectors from each source, accounting for direction. For charges of opposite sign, fields partially cancel; same sign, they add.

Frequently Asked Questions

What is an electric field and why is it useful? ▾
An electric field is a region where electric forces act on charged particles. Rather than thinking about charge-to-charge forces, the field concept says each charge creates a field that fills space, and other charges react to that field. This is useful because the field can be calculated once from source charges, then used to find forces on any test charge at any point.
How does E = kQ/r² compare to Coulomb's Law? ▾
Coulomb's Law: F = kQq/r². Electric field E = F/q = kQ/r². The field E is the force per unit test charge — a property of the source charge Q alone, independent of the test charge q. Once E is known, force on any charge: F = qE.
What is electric field inside a conductor? ▾
Inside a conductor in equilibrium, E = 0. Any external field redistributes surface charges until they create an equal and opposite internal field, perfectly canceling it. This is the Faraday cage effect — a conducting enclosure shields its interior from external electric fields.
What is the breakdown field strength of air? ▾
About 3×10⁶ V/m (3 MV/m). Above this, the field ionizes air molecules, creating a conductive plasma path. This is the mechanism of lightning, corona discharge on power lines, and spark plugs. SF₆ gas has a breakdown field of ~9 MV/m, allowing compact high-voltage equipment.
How does E relate to potential V? ▾
E = −∇V (the field is the negative gradient of potential). In 1D: E = −dV/dx. Between parallel plates: E = V/d (uniform field). High E means rapid change in V across space — the field 'pushes' charges from high V to low V.
What is Gauss's Law? ▾
Φ = Q_enclosed/ε₀ — the total electric flux through any closed surface equals the enclosed charge divided by ε₀. This powerful law simplifies E calculations for symmetric charge distributions (spheres, cylinders, planes). For a sphere of radius r enclosing Q: E × 4πr² = Q/ε₀ → E = Q/(4πε₀r²) = kQ/r² ✓
What is an electric dipole? ▾
A pair of equal and opposite charges ±q separated by distance d. The dipole moment p = qd. The field far from a dipole falls as 1/r³ (faster than 1/r² for a single charge). Dipoles are important in molecular chemistry, antenna theory, and microwave absorption.
How do electric fields relate to magnetism? ▾
By Maxwell's equations, a changing electric field creates a magnetic field and vice versa. This coupling propagates as electromagnetic waves at speed c = 1/√(ε₀μ₀). The relationship between E and B in an EM wave: E/B = c. Static E and B fields are independent; only changing fields link them.

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Interpretation: This formula links charge, voltage, current, resistance, capacitance, power or circuit time response. Assumption: Confirm DC versus AC conditions, RMS versus peak values, component topology and steady-state versus transient behavior. Ideal components may be assumed.

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