# What is the electric field?

Electric field calculator: E = kq ÷ r² of a point charge at a distance, with its direction, the potential and the Coulomb force on a second charge; or the distance or charge for a field.

- Page: https://www.acalculator.org/physics/electric-field-calculator
- JSON spec: https://www.acalculator.org/physics/electric-field-calculator.json
- Version: 7d18eb014661

## Default answer

Example with the default inputs (Find Electric field, Source charge (q) 2, Charge unit e (proton charges), Distance (r) 0.0000000000265 m, Second charge (Q) −1, Second charge unit e (proton charges)): E = 4.101 × 10¹² N/C.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| find | Find | Which value to work out from E = kq ÷ r². |
| q | Source charge (q) | The point charge that makes the field; negative for a negative charge. |
| qu | Charge unit | The unit of the charge. |
| r | Distance (r) | The distance from the charge to the point. |
| E | Field strength (E) | The size of the electric field at the point, in the unit chosen below. |
| Eu | Field unit | The unit of the field strength. |
| q2 | Second charge (Q) | Optional: a charge placed at the point, to find the Coulomb force on it. |
| q2u | Second charge unit | The unit of the charge. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The value worked out, to 6 significant figures. |
| field | Electric field (N/C) | E = k\|q\| ÷ r², in newtons per coulomb (the same as volts per metre). |
| direction | Field direction | Away from a positive charge, toward a negative one. |
| distance | Distance (m) | r, in metres. |
| charge | Charge size (C) | \|q\| = E r² ÷ k, in coulombs. |
| charges | Charge size (e) | \|q\| in proton charges, e = 1.602176634 × 10⁻¹⁹ C. |
| potential | Electric potential (V) | V = kq ÷ r, in volts, taking 0 far away. |
| force | Coulomb force (N) | F = k\|qQ\| ÷ r² = \|Q\|E, the size of the force on the second charge. |
| interaction | Force between the charges | Repulsive for charges of the same sign, attractive for opposite signs. |

## Method

E = k|q| ÷ r² with k = 1 ÷ (4πε₀); r = √(k|q| ÷ E); |q| = E r² ÷ k; V = kq ÷ r; F = k|qQ| ÷ r² = |Q| E.

## Assumptions

- A point charge (or a sphere of charge, seen from outside it) alone in a vacuum; ε₀ = 8.8541878188 × 10⁻¹² F/m.
- The second charge does not move the first; the potential is 0 far away.

## Worked examples

1. find = field, q = 2, qu = e, r = 0 gives E = 4,101,002,625,466.093, direction = Away from the charge, result = E = 4.101 × 10¹² N/C. Source: OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19).
2. find = field, q = 1, qu = e, r = 0, q2 = -1, q2u = e gives force = 0, interaction = Attractive: the charges pull together. Source: OpenStax, University Physics Volume 2, §5.3 Coulomb’s Law (Equation 5.1, F = (1 ÷ 4πε₀)|q₁q₂| ÷ r²; Example 5.1: the electron and proton of hydrogen at 5.29 × 10⁻¹¹ m attract with 8.25 × 10⁻⁸ N), https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law (retrieved 2026-10-19) (8.25 × 10⁻⁸ N with k = 8.99 × 10⁹ and e = 1.602 × 10⁻¹⁹ C); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19).
3. find = field, q = 1, qu = nC, r = 1 gives E = 8.987552, potential = 8.987552, result = E = 8.98755 N/C. Source: OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19).
4. find = distance, q = 1, qu = uC, E = 10,000, Eu = NC gives r = 0.948027, result = r = 0.948027 m. Source: OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19).
5. find = charge, E = 10,000, Eu = NC, r = 1 gives charge = 0.000001. Source: OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19).

## FAQ

### How do you calculate the electric field of a point charge?

E = kq ÷ r², with k = 1 ÷ (4πε₀) = 8.98755 × 10⁹ N·m²/C², q the charge and r the distance. A 1 nC charge makes E = 8.98755 × 10⁹ × 10⁻⁹ ÷ 1² = 8.99 N/C at 1 m.

### Which way does the electric field point?

Away from a positive charge and toward a negative one: the field shows the push on a small positive test charge. The field of the helium nucleus (+2e) points away from it.

### How is the electric field related to Coulomb’s law?

Coulomb’s law gives the force between two charges, F = k|qQ| ÷ r². Divide by the second charge and you get the field of the first, E = F ÷ Q. So a charge Q in a field E feels F = QE.

### What is the force between a proton and an electron in hydrogen?

With e = 1.602176634 × 10⁻¹⁹ C and r = 5.29 × 10⁻¹¹ m, F = k e² ÷ r² = 8.24 × 10⁻⁸ N, attractive. OpenStax gets 8.25 × 10⁻⁸ N with rounded constants.

### What are the units of electric field?

Newtons per coulomb (N/C), which is the same as volts per metre (V/m). Large fields are given in kV/m or MV/m.

### How does the field change with distance?

It falls with the square of the distance: twice as far, a quarter of the field; ten times as far, one hundredth. The potential, V = kq ÷ r, falls only with the distance itself.

## Sources

- OpenStax, University Physics Volume 2, §5.4 Electric Field: the field of a point charge E = (1 ÷ 4πε₀)(q ÷ r²) r̂; Equation 5.3, F = QE; Example 5.3 (the field of a helium nucleus, charge 2e, at 26.5 × 10⁻¹² m: 4.1 × 10¹² N/C, pointing away), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field
- OpenStax, University Physics Volume 2, §5.3 Coulomb’s Law: Equation 5.1, F = (1 ÷ 4πε₀)|q₁q₂| ÷ r²; Example 5.1 (the proton and electron of hydrogen at 5.29 × 10⁻¹¹ m: 8.25 × 10⁻⁸ N, attractive), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law
- OpenStax, University Physics Volume 2, §7.3 Calculations of Electric Potential: Equation 7.8, V = kq ÷ r for a point charge, with V = 0 at infinity, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/7-3-calculations-of-electric-potential
- NIST, CODATA 2022: vacuum electric permittivity ε₀ = 8.8541878188(14) × 10⁻¹² F/m, retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?ep0
- NIST, CODATA 2022: elementary charge e = 1.602176634 × 10⁻¹⁹ C (exact), retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?e