Electric Field Calculator

What an electric field measures

An electric field describes the push or pull that a charged object would feel at a point in space, before any charge is placed there. Put a small positive test charge at the point, measure the force on it, and divide that force by the size of the test charge. The result is the field strength at that point. The unit is newtons per coulomb, written N/C, and it is identical to volts per meter, so both labels appear in textbooks and on instrument dials.

The idea separates two things that are easy to blur. A source charge creates a field everywhere around it. A second charge placed in that field feels a force, and the force depends on both the field and the size of the second charge. The field itself depends only on the source and on where you stand. That is why a field map can be drawn once and reused for any particle you later send through it.

Field is a vector, so it has a direction as well as a size. Around a positive source charge the field points straight away from the charge. Around a negative source charge it points straight in. A positive test charge released from rest moves along the field line in the direction of the arrow, and a negative one moves against it.

The point charge formula

For a single point charge in vacuum, the field strength at a distance r is:

E = k x |q| / r^2

Here q is the source charge in coulombs, r is the distance from the charge to the point in meters, and k is Coulomb's constant, 8.9875517923 x 10^9 N m^2/C^2. The absolute value keeps the magnitude positive; the sign of q is used for the direction instead. The value of k comes from the vacuum permittivity, and the NIST CODATA constants list the underlying quantities with their uncertainties.

Two features of the formula drive almost every result on this page. The charge sits in the numerator, so doubling the charge doubles the field. The distance sits in the denominator and is squared, so doubling the distance cuts the field to one quarter. Neither rule is optional, and most wrong answers come from breaking one of them by entering a distance in centimeters or by squaring the charge by mistake.

Strictly the formula describes a point charge, or the field outside a uniformly charged sphere, which behaves as if all its charge sat at the center. Close to an extended object, or inside a conductor, the field has a different shape and this equation no longer applies. The MIT notes on Coulomb's law derive the point charge field from the force law if you want the full step by step.

Worked example: 2 microcoulombs at 30 centimeters

Take a source charge of 2 microcoulombs and a point 30 centimeters away. Convert first: 2 microcoulombs is 2 x 10^-6 C, and 30 centimeters is 0.30 m. The distance squared is 0.09 m^2.

E = 8.9875517923 x 10^9 x 2 x 10^-6 / 0.09. The numerator works out to 17,975.10, so 17,975.10 / 0.09 = 199,723.37 N/C. That is roughly 2.00 x 10^5 N/C, a strong field by everyday standards but ordinary for a laboratory demonstration.

The inverse square rule gives two more points without touching the calculator. At 60 centimeters, twice the distance, the field is one quarter as large, or 49,930.84 N/C. At 15 centimeters, half the distance, it is four times as large, or 798,893.49 N/C. Moving from 15 to 60 centimeters is a factor of four in distance and a factor of sixteen in field, which is why the numbers change so quickly near the charge.

A 2 microcoulomb charge is small in absolute terms. One coulomb is the charge carried by roughly 6.24 x 10^18 electrons, so a microcoulomb is a few trillion electrons. The field is large because those charges are packed into a small object and the constant k is large in SI units.

How distance and charge move the number

The table below holds the charge at 2 microcoulombs and walks the distance outward. Every row uses the same formula, so the pattern is the point: each doubling of distance divides the field by four.

Charge works on a straight line instead. At a fixed 0.30 m, a 1 nanocoulomb charge gives 99.86 N/C, a 100 nanocoulomb charge gives 9,986.17 N/C, and a 10 microcoulomb charge gives 998,616.87 N/C. Ten thousand times the charge gives ten thousand times the field, because nothing in the formula raises q to a power.

Read the two tables together and you can estimate a field without a calculator. Pick the nearest row, then scale by the ratio of charges and the square of the ratio of distances. A 3 microcoulomb charge at 0.45 m is half the 2 microcoulomb row at 0.30 m multiplied by 1.5, so about 149,793 N/C, which is close to the exact 133,149.02 N/C and good enough to catch a unit error.

Field, force and the test charge

Once the field is known, the force on any charge placed at that point is F = q x E. A 1 nanocoulomb test charge sitting in the 199,723.37 N/C field from the first example feels 199,723.37 x 1 x 10^-9, or 1.997 x 10^-4 N. That is about 0.2 millinewtons, small enough that the test charge barely disturbs the source, which is exactly what a test charge is supposed to do.

Swap the roles and the same equation recovers the field. If a 5 nanocoulomb charge feels 2.25 x 10^-4 N, the field at its location is 2.25 x 10^-4 / 5 x 10^-9, or 45,000 N/C. This route is useful when the source is hidden inside an enclosure and only the force on a known probe can be measured.

Field strength and force differ in one more way that matters for signs. The force on a positive charge points along the field, and the force on a negative charge points against it. An electron in a field pointing east accelerates west. The magnitude of the force is the same for both signs if the charge sizes match, so a field of 199,723.37 N/C pushes a 1 nanocoulomb charge and pulls a 1 nanocoulomb negative charge with 1.997 x 10^-4 N in opposite directions.

Direction, signs and adding fields

Direction is set by the source charge, not by the formula. A positive source pushes a positive test charge away, so the field points outward along the line from the source. A negative source pulls a positive test charge in, so the field points inward toward the source. At a point 30 centimeters to the right of a negative charge, the field arrow points left.

With several source charges, the fields add as vectors. Two equal positive charges have a point midway between them where the two field arrows are equal and opposite, so the total field there is zero. Move closer to one charge and the nearer field wins. Two opposite charges of equal size add instead of cancel along the line joining them, because both fields point the same way in that region.

Vector addition means adding components, not sizes. Two fields of 100 N/C at right angles give 141.42 N/C, not 200. Two fields of 100 N/C pointing in opposite directions give zero. The calculator on this page handles one source charge at a time; for a pair, run it once per charge and add the x and y components by hand.

Where field calculations matter in the real world

Electrostatic shielding is the best known application. A Faraday cage is a hollow metal enclosure, and a charge placed on the outside surface rearranges itself so the interior stays at a constant potential with zero strength inside. Cars and aircraft act as rough cages during lightning storms, which is why safety guidance tells people to shelter inside a vehicle rather than under a tree. Coaxial cable, shielded microphone wire and even a microwave oven door all exploit the same principle.

Industrial equipment leans on the inverse square behaviour too. An electrostatic precipitator charges smoke and dust particles, then pulls them onto collector plates, scrubbing chimney exhaust at power plants. Photocopiers charge a drum so toner sticks only where a laser has discharged it. Automotive painters spray paint through an electrostatic gun so droplets wrap around body panels and overspray drops. High voltage engineers compute the field near transmission line hardware because air breaks down and sparks once the strength crosses roughly 3 megavolts per metre at standard humidity, which sets minimum clearance distances around substation equipment.

Medical and laboratory instruments depend on the same arithmetic. An electrophoresis gel separates DNA fragments by driving them through a gel with an applied potential difference, and the migration speed tracks the strength between the electrodes. Bioelectric measurements such as electrocardiograms read millivolt signals produced by heart muscle, so instrument designers shield the leads to keep stray pickup out. Physics labs generate intense fields with a Van de Graaff generator, whose belt carries charge onto a hollow dome until corona discharge leaks it away. Electrostatic headphones push a thin diaphragm between two perforated stators, and inkjet printers steer charged droplets past deflection plates. Semiconductor fab equipment, dust sensors, ozone generators and laser printers all trace their design curves back to the point charge expression this page evaluates.

Scanning tunneling microscopes bring a sharpened probe within a nanometer of a wafer surface, and the tunneling current that flows depends on the gap, letting the instrument map individual atoms. Electrostatic chucks clamp silicon discs inside vacuum etch chambers, ionizers neutralize static on film lines, bug zappers lure insects toward an electrified grid, and air purifiers charge smoke particles so plates downstream catch them. Ranchers use electrified fence energizers, grain elevators monitor dust hazards, and paint booths measure grounding before spraying flammable solvents. Older display tubes made the principle visible: a hot cathode in a cathode ray tube boiled electrons off, deflection plates steered the beam across phosphor lines, and a mass spectrometer bends ion trajectories the same way to sort isotopes by their charge to mass ratio.

Electrostatic loudspeaker drivers, Wimshurst machines and classroom pith ball demonstrations round out the household list, and museum staff keep a Van de Graaff dome behind rope so visitors can try the standing hair trick without touching the charging belt. Farmers even rely on electrostatic seed coating and crop sorting rigs, where chaff separates by how strongly each kernel polarizes.

Units, prefixes and common mistakes

Electric field is measured in newtons per coulomb or volts per meter, and the two are the same unit written differently. One volt is one joule per coulomb, and one joule per meter is one newton, so V/m reduces to N/C. A field of 1,000 V/m is 1,000 N/C. Engineers often quote kilovolts per meter, where 1 kV/m equals 1,000 V/m, and breakdown strengths for air are usually given in that unit.

Four mistakes account for most wrong answers. Entering centimeters as meters makes the field 10,000 times too small, because the error is squared. Squaring the charge instead of the distance inverts the scaling. Using a diameter where a radius belongs doubles the distance and quarters the field. And treating a field as constant across a region ignores the inverse square drop, which is large once the distance changes by more than a few percent.

A quick order of magnitude check catches all four. A nanocoulomb at a meter should give about 9 N/C, because k is about 9 x 10^9 and the distance squared is 1. A microcoulomb at the same meter should give about 9,000 N/C. If your answer is off by a factor of a hundred or more, the units are the first place to look.