Point Charge Electric Field Calculator

Definition of the field of a point charge

The electric field at a point is the force that would act on a positive test charge of 1 C placed there, if that test charge did not disturb the source. For a charge concentrated at one point, in vacuum, the field at distance r is E = k·Q/r². The direction is radial. A positive Q repels the test charge, and a negative Q attracts it. This calculator returns a number with the sign of Q: a positive result points away from the charge, and a negative result points toward it.

The reference values on this page are Q = 1 µC = 1×10⁻⁶ C and r = 0.1 m, with the k field left blank. The calculator uses k = 8.9875517923×10⁹ N·m²/C². The product k·Q is 8987.5517923 N·m²/C, r² is 0.01 m², so E is 898,755.17923 N/C. The panel shows 898,755.1792 N/C. That pair of inputs comes back in the later examples. The unit N/C equals V/m.

History: Coulomb, Cavendish, and the field

Charles-Augustin de Coulomb published, in 1785 in the Histoire de l’Académie royale des sciences, measurements of the force between charges on a torsion balance. The force fell as the inverse square of the distance. Henry Cavendish had reached the same law in experiments in the 1770s. James Clerk Maxwell published Cavendish’s results only in 1879, in The Electrical Researches of the Honourable Henry Cavendish.

Coulomb measured a force, not a field vector. Michael Faraday developed the field as a state of space in the 1830s and 1840s, and Maxwell gave it a mathematical form. Today’s formula E = k·Q/r² is Coulomb’s force divided by the test charge. The constant k is 1/(4πε₀). The number stored in the calculator, 8.9875517923×10⁹, is a widely printed approximation of that constant, close to c²×10⁻⁷ from the era of an exact μ₀. It is not a live CODATA reading after the 2019 SI redefinition, when ε₀ became a measured quantity. School problems often use 9×10⁹. Enter that number in the k field.

Derivation of E = k·Q/r²

Step 1. Coulomb’s law: the force between charges Q and q separated by r is F = k·Q·q/r², along the line that joins them. Step 2. The field is defined as the limit of F/q when the test charge q is small enough not to move the source. Step 3. Dividing the force by q cancels q and leaves E = k·Q/r². The test charge is therefore not an input of this calculator.

Step 4. For the reference values, r² = (0.1)² = 0.01 m², and k·Q = 8.9875517923×10⁹ × 10⁻⁶ = 8987.5517923. The quotient is 898,755.17923 N/C, which the panel prints as 898,755.1792 N/C. With the school k = 9×10⁹ the same charge and the same distance give 900,000.0000 N/C. The difference comes only from the constant, not from a different formula.

Assumptions: a point charge, vacuum, and a static field

The calculator assumes one charge concentrated at a point, vacuum, and an electrostatic field. It does not divide k by the relative permittivity of a medium. If the charge sits in a dielectric, enter an already reduced constant. The real ball must be small compared with r. At the surface of a sphere of finite radius the point model overstates the field, because the charge is not at a single point.

The calculator does not compute the field of an infinite line, a plane, a dipole, or a parallel-plate capacitor. It does not add fields from several charges, because superposition is a vector sum and the result here is one signed scalar. It does not compute the potential V = k·Q/r. At r = 0 the formula is singular, and the calculator refuses to divide. Moving charges and the magnetic field lie outside this model.

The sign of the charge and the inverse-square law

The sign of E is the sign of Q. For Q = −2 µC and r = 0.5 m the result is -71,900.4143 N/C, and the absolute value is 71,900.4143 N/C. Doubling the distance divides the field by four, not by two. From 0.1 m to 0.2 m the reference value falls from 898,755.1792 N/C to 224,688.7948 N/C. Moving in from 0.1 m to 0.05 m gives 3.5950e+6 N/C, because when |E| is at least 10⁶ the panel uses scientific notation.

The force on a known test charge is F = q·E, but this calculator has no field for q. The definition E = F/q has its own calculator, and the force between two charges is covered by the Coulomb force calculator. Potential and capacitor capacitance also have their own calculators, because those formulas take different inputs.

Units, magnitudes, and what the model is for

The coulomb, the newton, and the volt are SI units, as in the BIPM brochure. One N/C equals 1 V/m. Enter the charge in coulombs: 1 µC is 1×10⁻⁶ C, and 1 nC is 1×10⁻⁹ C. Enter the distance in meters. With US units selected in the header, the r row shows feet and the meter value in parentheses, while k and Q stay in SI. A comma and a period in the fields are equivalent.

The figures below are results of the formula for the stated inputs, not a laboratory measurement. A 1 µC ball at 10 cm gives 898,755.1792 N/C. The same ball at 1 m gives 8,987.5518 N/C. A 100 nC model at 1 cm, treated as a point, gives 8.9876e+6 N/C. A charge of 10 µC at 30 cm gives 998,616.8658 N/C. A charge of 2 µC at 2 m gives 4,493.7759 N/C. The fair-weather field near the ground is often on the order of 100 V/m, but it comes from clouds and the ground, not from one point charge, so this calculator does not describe it.

Sources

The definition, the derivation, and the history come from primary accounts and textbooks: C.-A. de Coulomb, memoirs in the Histoire de l’Académie royale des sciences, 1785; J. C. Maxwell (ed.), The Electrical Researches of the Honourable Henry Cavendish, Cambridge 1879; D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, the chapter on the field of a point charge; H. D. Young and R. A. Freedman, University Physics, the chapter on the field of a point charge; D. J. Griffiths, Introduction to Electrodynamics, chapter 2; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume II, chapter 4 (Electrostatics); BIPM, SI Brochure, 9th edition; NIST, CODATA electric constant. This page is for educational purposes. The constant in the code is not a live CODATA reading.