Coulomb's Law and the Electric Field of a Point Charge

Q: What is the difference between k and ε₀?

They are the same information written two ways: k = 1/4πε0 = 8.987551786×109 N·m²/C². Indian board papers usually write the law with 1/4πε0 in front; Cambridge 9702 writes F = Q₁Q₂/(4πε0r²) with the constant in the denominator. Neither is a different physics. If a paper hands you ε0 = 8.85×10−12 and you want k, compute 1/(4π × 8.85×10−12).

Q: Does the force change if I put the charges in water instead of air?

Yes, and by a lot: it falls by the factor εr, which for water at room temperature is of the order of 80. That is one of the reasons ionic compounds dissolve in water and not in petrol — the attraction holding the ions together is screened to about one eightieth of its vacuum value. Enter the relative permittivity in the fourth field. This page deliberately does not offer a dropdown of materials, because the values are measured properties that would need a sourced table and none could be sourced for this batch.

Q: How many electrons is a microcoulomb?

Divide by the elementary charge: 10−6 / 1.602176634×10−19 = 6.24×1012 electrons. The calculator prints this for q₁. It is worth doing once, because it shows how small a charge a microcoulomb is in atomic terms and how enormous in mechanical ones: six million million electrons, held 10 cm from another lot, pushing with the weight of half a kilogram.

Give me two charges and how far apart they are, and this returns the force on each of them, the field each one makes where the other sits, the signed answer that separates attraction from repulsion, the pair’s potential energy, and the separation that would make the force exactly one newton.

Last updated 11 Nov 2026

Coulomb force and point-charge field

Two charges → force, field, energy

Charge q₁, in microcoulombs (µC)Type the sign. A microcoulomb is 10⁻⁶ C and is about as large as a charge gets in a school question — a van de Graaff dome holds a few of them. If your question gives nanocoulombs, divide by 1000; if it gives the charge as a number of electrons, multiply by 1.602176634 and enter the answer in units of 10⁻¹³ µC, or use the elementary-charge row below to check yourself.

Charge q₂, in microcoulombs (µC)Type the sign here too. The headline force is a MAGNITUDE and so is always positive; the signed row beneath it is the one that tells you attraction from repulsion, and it is negative exactly when the two charges have opposite signs.

Separation between the charges, in centimetresCentre to centre. Coulomb’s law is a law about POINT charges: it is exact for two points, exact outside two uniformly charged spheres if you measure between their centres, and wrong for two charged conductors close enough to polarise each other. As a rough working rule, treat spheres as points once they are several diameters apart.

Relative permittivity of the medium, εᵣLeave this at 1 unless the question says otherwise — 1 is vacuum, exactly and by definition, and dry air differs from it by about six parts in ten thousand, which is far below the precision of any school answer. Water is of the order of 80 at room temperature, which is why ionic solids come apart in it: the same two ions attract each other some eighty times less strongly. This page does not print a table of dielectric constants, because a table of them is a table of measured material properties and none of the values could be sourced to a publication for this batch.

5.3925NExample

A 2 µC charge and a −3 µC charge, 10 cm apart in air

One inverse square, one sign rule, and a constant that is no longer exact

F = (1/4πε₀ε_r) · |q₁q₂| / r² · E = (1/4πε₀ε_r) · |q| / r² · U = (1/4πε₀ε_r) · q₁q₂ / r

q₁, q₂: the two charges, WITH their signs. The product q₁q₂ is what carries the sign of the answer: negative product means opposite charges means attraction. Put both in coulombs before you multiply — a microcoulomb left as “2” instead of 2×10⁻⁶ puts the force out by a factor of 10¹², which is the single commonest arithmetic failure on this topic
r: the separation, in METRES, and squared. Centimetres left unconverted put the force out by exactly 10⁴; millimetres by 10⁶. The square is what makes the error so large, and it is also what makes the answer sensitive: 10 per cent out on r is 21 per cent out on F
1/4πε₀: 8.987 551 786 × 10⁹ N·m²/C², often rounded to 9×10⁹ in a school answer, which costs 0.14 per cent. Note what it is NOT: an exact number. Since 2019 ε₀ is a measured quantity with an uncertainty of about 1.6 parts in 10¹⁰, so this constant has one too
εᵣ: the relative permittivity of whatever the charges sit in — 1 for vacuum by definition, near enough 1 for air, of order 80 for water. It DIVIDES the force. A medium never increases the Coulomb force between two charges; it screens them
E = F/q: the field. Divide the force by the charge that feels it and what is left belongs to the source alone. In newtons per coulomb, which is the same unit as volts per metre — the two are identical, and a question may use either

Worked example

A 2 µC charge and a −3 µC charge, 10 cm apart in air
CONVERT FIRST, BEFORE ANYTHING ELSE. q₁ = 2×10⁻⁶ C, q₂ = −3×10⁻⁶ C, r = 0.10 m. Air is close enough to vacuum that ε_r = 1. Everything that goes wrong on this topic goes wrong here, not in the physics
THE PRODUCT, WITH ITS SIGN. q₁q₂ = (2×10⁻⁶)(−3×10⁻⁶) = −6×10⁻¹² C². The minus sign is the whole answer to “attract or repel” and you have it before you touch the constant
DIVIDE BY r². r² = 0.010 m², so q₁q₂/r² = −6×10⁻¹⁰ C²/m²
MULTIPLY BY THE CONSTANT. 8.987551786×10⁹ × (−6×10⁻¹⁰) = −5.3925 N. The magnitude is 5.3925 N and the sign says ATTRACTION. Rounding the constant to 9×10⁹ would have given 5.4 N, which is the same answer to two figures
SANITY-CHECK IT AGAINST SOMETHING YOU KNOW. 5.39 N is about 550 times the weight of a one-gram mass, from two charges you could not see, 10 cm apart. That is the real lesson of Coulomb's law: the electrostatic force is enormous compared with gravity at everyday scales, and the only reason you do not notice it is that matter is almost exactly neutral
AND CHECK THE FIELD SEPARATELY. q₁ alone makes a field of 1,797,510 V/m where q₂ sits. Multiply that by 3×10⁻⁶ C and you get 5.3925 N again — the same force, computed the other way round, which is what F = qE means

Which quantity falls how fast, for one point charge

Two of these fall as the square and two as the first power, and mixing them up is a standard way to lose a mark. The reason is in the units: a field is a force per charge and a potential is an ENERGY per charge, and energy is force times distance, so the potential always carries one less power of r than the field does. If you remember only one thing from this table, remember that E and V do not fall off at the same rate.
Quantity	Formula	Falls as	At twice the distance	At ten times
Force between two charges	kq₁q₂/r²	1/r²	× 0.25	× 0.01
Field of one charge	kq/r²	1/r²	× 0.25	× 0.01
Potential of one charge	kq/r	1/r	× 0.5	× 0.1
Potential energy of a pair	kq₁q₂/r	1/r	× 0.5	× 0.1

What the sign of each printed quantity actually means

The potential energy row is the one worth stopping on. A negative U is not a smaller energy than a positive one in some loose sense — it means the pair is BOUND, and the number is exactly the energy you would have to supply to pull them infinitely far apart. The zero is at infinity by convention, which is a choice, not a fact; nothing measurable depends on where the zero goes, only on differences.
Printed row	Negative means	Positive means
Headline force	never negative — it is a magnitude	always
Signed force	the charges attract (opposite signs)	the charges repel (same signs)
Potential energy of the pair	bound: you must DO work to separate them	you would have to do work to bring them together, and they fly apart if released
Net field at the midpoint	points from q₂ towards q₁	points from q₁ towards q₂

Why the sign, the unit conversion and the medium matter more than the formula

Coulomb’s law is one line, so the useful part of a page about it is everything around the line. Three things cost marks, and none of them is the formula.

The first is the unit conversion, and it is not close. Charges arrive in microcoulombs or nanocoulombs and separations in centimetres or millimetres, while the constant is in metres and coulombs. A microcoulomb entered as a plain number is a factor of 10⁶ per charge; a separation in centimetres is a factor of 10⁴ because it is squared. Convert both before you touch the constant, every single time.

The second is the sign. The magnitude of the force uses the magnitudes of the charges, and the DIRECTION comes from the sign of the product. Carry the signs into the product and the answer tells you attraction or repulsion for free; drop them and you have to reason it out afterwards, which is where it goes wrong under exam pressure. This page prints both the magnitude and the signed value for that reason. It also prints the pair’s potential energy, which is signed in the same way but falls as 1/r rather than 1/r² — see electric potential and potential energy for what the sign of that one is telling you.

The third is that the field is not the force. E = F/q looks like a rearrangement and is really a change of subject: the field is what the source charge does to the space around it, whether or not anything is there to feel it. That is why a field has a value at a point with no charge in it, and it is the idea the whole of electrostatics is built on. A charge placed in a field of 1,797,510 V/m feels 1.8 million newtons per coulomb — which sounds absurd until you remember that nobody has a coulomb.

And one thing that is genuinely new. Since 20 May 2019 the elementary charge is exact BY DEFINITION — 1.602176634×10⁻¹⁹ C, no uncertainty, because the coulomb is now defined from it. What lost its exactness in the same revision is ε₀, which used to be exactly 1/μ₀c² and is now a measured quantity: 8.8541878188(14)×10⁻¹² F/m. So the constant in Coulomb’s law has an uncertainty and the charge of the electron does not, which is the exact reverse of what almost every school textbook still implies.

Frequently asked questions

Do I use the signs of the charges in the formula or not?

Both, in different places. For the MAGNITUDE of the force, use the magnitudes: F = k|q₁q₂|/r². For the direction, use the sign of the product q₁q₂: negative means they attract, positive means they repel. The tidiest habit is to keep the signs all the way through, read the sign of the answer as the direction, and then quote the magnitude — which is what this page prints on two separate rows so that neither reading can be mistaken for the other.

What is the difference between k and ε₀?

They are the same information written two ways: k = 1/4πε₀ = 8.987551786×10⁹ N·m²/C². Indian board papers usually write the law with 1/4πε₀ in front; Cambridge 9702 writes F = Q₁Q₂/(4πε₀r²) with the constant in the denominator. Neither is a different physics. If a paper hands you ε₀ = 8.85×10⁻¹² and you want k, compute 1/(4π × 8.85×10⁻¹²).

Why is the force the same on both charges when one of them is much bigger?

Newton’s third law, and the arithmetic shows it: the formula contains the PRODUCT q₁q₂, which is the same number whichever charge you call the source. A 1 µC charge and a 1000 µC charge pull on each other with identical force. Their ACCELERATIONS differ, because those depend on mass, and that is usually what a question is really asking about when it seems to be asking about force.

Does the force change if I put the charges in water instead of air?

Yes, and by a lot: it falls by the factor ε_r, which for water at room temperature is of the order of 80. That is one of the reasons ionic compounds dissolve in water and not in petrol — the attraction holding the ions together is screened to about one eightieth of its vacuum value. Enter the relative permittivity in the fourth field. This page deliberately does not offer a dropdown of materials, because the values are measured properties that would need a sourced table and none could be sourced for this batch.

How many electrons is a microcoulomb?

Divide by the elementary charge: 10⁻⁶ / 1.602176634×10⁻¹⁹ = 6.24×10¹² electrons. The calculator prints this for q₁. It is worth doing once, because it shows how small a charge a microcoulomb is in atomic terms and how enormous in mechanical ones: six million million electrons, held 10 cm from another lot, pushing with the weight of half a kilogram.

Can I use Coulomb’s law for two charged metal spheres?

Outside two uniformly charged spheres the law is exact if you measure r between their CENTRES — that is a standard result from Gauss’s theorem, which the CBSE Unit I list covers. The catch is the word uniformly. Bring two conductors close together and each one’s charge redistributes under the other’s field, so neither is uniform any more and the simple formula drifts. Several diameters apart, it is fine.

Related calculators

References

2022 CODATA Recommended Values of the Fundamental Physical Constants (wallet card), NIST Committee on Data for Science and Technology, physics.nist.gov/cuu/pdf/wallet_2022.pdf (read 11 November 2026). Every constant on this page is taken from here, and the reason to read the card rather than a textbook is the bracket. The elementary charge is printed as 1.602 176 634 × 10⁻¹⁹ C with no bracket at all, because since the 2019 revision of the SI it is exact by definition — the coulomb is now defined from it, not the other way round. The same is true of c, h, k and N_A. But the vacuum electric permittivity is printed 8.854 187 8188(14) × 10⁻¹² F/m and the vacuum magnetic permeability 1.256 637 061 27(20) × 10⁻⁶ N/A²: both now have an uncertainty, about 1.6 parts in 10¹⁰, where before 2019 μ₀ was exactly 4π × 10⁻⁷ and ε₀ exactly 1/μ₀c². Almost every school book still prints the old exact forms. At school precision nothing changes; the logic of which constants are definitions changed completely.
Annual Syllabus Session 2023-24, Class XII Physics (Code 042), Central Board of Secondary Education, edustud.nic.in/edu/SYLLABUS_2023_24/12/12_physics_eng_2023_24.pdf (read 11 November 2026). The board’s own document, cited and not endorsed by it. Unit I Chapter 1 asks for “Coulomb’s law-force between two point charges; forces between multiple charges, superposition principle”; Chapter 2 for “electrical potential energy of a system of two point charges” and “capacitance of a parallel plate capacitor with and without dielectric medium between the plates, energy stored in a capacitor.(No derivation, formulae only)”. Unit II asks for “drift velocity, mobility and their relation with electric current”, “Internal resistance of a cell, potential difference and emf of a cell, combination of cells in series and in parallel, kirchhoff’s rules, Wheatstone bridge”. Unit III asks for “Biot – Savart law and its application to the current carrying circular loop”, “Ampere’s law and its applications to infinitely long straight wire. Straight solenoid (only qualitative treatment)” and “force between two parallel current-carrying conductors-definition of ampere”.
Syllabus: Cambridge International AS & A Level Physics 9702, for examination in 2025, 2026 and 2027 (Version 1), Cambridge Assessment International Education, cambridgeinternational.org/Images/664565-2025-2027-syllabus.pdf (read 11 November 2026). Cited for the outcomes a Cambridge candidate is examined on, which are worded differently from the Indian board’s: 18.3.2 “recall and use Coulomb’s law F = Q1Q2 /(4πε0r2) for the force between two point charges in free space”, 18.5.3 “use V = Q/(4πε0r) for the electric potential in the field due to a point charge”, 9.1.4 “use, for a current-carrying conductor, the expression I = Anvq, where n is the number density of charge carriers”, 10.1.5 “understand the effects of the internal resistance of a source of e.m.f. on the terminal potential difference”, 19.1.3 “derive, using C = Q/V, formulae for the combined capacitance of capacitors in series and in parallel”, 20.2.2 “recall and use the equation F = BIL sin θ, with directions as interpreted by Fleming’s left-hand rule” and 20.3.2 “recall and use F = BQv sin θ”. The copy read here was truncated before the data-and-formulae sheet, so what is on THAT sheet could not be established from it.
The International System of Units (SI), 9th edition, Bureau International des Poids et Mesures, 2019. Cited for the structure the 2019 revision imposed: the ampere is now fixed by taking the elementary charge to be exactly 1.602 176 634 × 10⁻¹⁹ C, so the coulomb is one ampere-second and the ampere is defined through a charge. It is no longer defined by the force between two parallel wires — that force is now a measured consequence, with μ₀‘s uncertainty in it. Several school syllabuses, the CBSE document cited above among them, still carry “definition of ampere” beside the two-wire force. The old definition is a good way to SEE what an ampere is; it has not been the definition since 20 May 2019.