Coulomb's Law Calculator

What Coulomb's law measures

Run a comb through dry hair and it will pick up bits of paper. Pull two freshly laundered socks apart and they fight you for a split second. Those pulls and pushes come from electric charge, and the force between two charges has a size you can predict with one formula. French physicist Charles-Augustin de Coulomb worked out the rule in 1785 with a torsion balance, and it has held up through every measurement since.

Coulomb's law says the force between two point charges grows with the size of either charge and shrinks with the square of the distance between them. Small everyday objects pick up charges measured in microcoulombs, millionths of a coulomb, yet the forces still reach tenths of a newton when the objects are close. That is enough to feel with your fingers and enough to yank a styrofoam peanut across a table.

The law treats each charged object as a single point with all its charge packed at the center. That sounds like a cheat, but it works well whenever the objects are small compared with the gap between them, which covers most classroom and household cases. The NIST CODATA fundamental constants list fixes the one number the formula needs, the Coulomb constant, to 11 significant digits.

The formula and what each letter means

The formula is F = k q1 q2 / r2. F is the force in newtons, q1 and q2 are the two charges in coulombs, r is the distance between them in meters, and k is the Coulomb constant, 8.9875517923 x 10^9 N m2/C2 by the NIST CODATA 2018 adjustment. For homework, 8.99 x 10^9 rounds close enough that the difference almost never shows up in the answer.

Charges rarely arrive in plain coulombs. A swept balloon carries around a tenth of a microcoulomb, and a doorknob spark involves a fraction of that. One microcoulomb, written µC, is 0.000001 C. One nanocoulomb, written nC, is 0.000000001 C. Move the decimal six places for µC and nine places for nC, and the rest of the calculation stays the same.

The unit itself has a modern definition. Since the 2019 revision of the SI system, described in the BIPM SI Brochure, the elementary charge is exactly 1.602176634 x 10^-19 C, and one coulomb is the charge of about 6.24 x 10^18 electrons. Every charge you will ever enter is a whole-number multiple of that tiny amount.

Worked example: two balloons 8 cm apart

Say you rub two balloons on a wool sweater and each ends up charged, one at 0.40 µC and one at 0.25 µC. You hang them 8.0 cm apart. First convert: 0.40 µC is 4.0 x 10^-7 C and 0.25 µC is 2.5 x 10^-7 C, while 8.0 cm is 0.080 m.

Now follow the formula. Multiply the charges: 4.0 x 10^-7 times 2.5 x 10^-7 is 1.0 x 10^-13. Multiply by k: 8.9875517923 x 10^9 times 1.0 x 10^-13 is 8.9876 x 10^-4. Square the distance: 0.080 squared is 0.0064 m2. Divide: 8.9876 x 10^-4 over 0.0064 comes out to 0.1404 N.

Both balloons carry the same sign of charge after a rub on wool, so the force pushes them apart. Fourteen hundredths of a newton sounds small, but it is about the weight of 3 sheets of paper, at 14 grams a sheet, which is why the balloons visibly swing away from each other.

Textbook examples you can check

OpenStax's College Physics 2e, section 18.3 works the electron and proton in a hydrogen atom. The separation is 0.530 x 10^-10 m and each charge is 1.60 x 10^-19 C in magnitude. With k = 8.99 x 10^9, the force works out to 8.19 x 10^-8 N, and the book reports the same figure. The pair attracts, since the charges have opposite signs.

The same book family shows how Coulomb measured the law in the first place. In OpenStax Physics, section 18.2, two spheres start 5.0 cm apart with a force of 20 x 10^-6 N between them. Bring them to 3.0 cm and the force becomes 20 x 10^-6 multiplied by (0.050 / 0.030) squared, which is 5.6 x 10^-5 N. No charge value is needed, because the ratio of distances does all the work.

Try the hydrogen numbers in the calculator with femtometer-level distances and the output should land on the same order of magnitude. That kind of cross-check is the fastest way to catch a unit slip before it costs you points on a test.

The inverse square part in plain numbers

The r2 at the bottom of the formula is the part people underestimate. Because distance is squared, small changes in spacing produce big swings in force. Double the distance and the force falls to one quarter. Triple it and the force falls to one ninth. Cut the distance in half and the force jumps to four times what it was.

The pattern runs in both directions, which is what makes the law handy in reverse. OpenStax's exercise set asks what happens to a 5.00 N force when the distance triples: 5.00 divided by 9 is 0.56 N, and no charge values are needed at all. The calculator on this page shows the halved and doubled distance rows for any input pair, so you can see the swing without doing the arithmetic.

Watch the units whenever distances get small. A gap of 2 cm is 0.02 m, and its square is 0.0004 m2. Enter 2 instead of 0.02 and the answer comes out 10,000 times too small, an error that is easy to make and easy to spot with the halved distance row.

Attract or repel: what the signs mean

Two charges with the same sign push apart, and two charges with opposite signs pull together. Electrons carry a negative charge and protons carry a positive one, so an electron and a proton attract while two electrons repel. The formula handles this automatically: if q1 times q2 is positive the force is repulsive, and if the product is negative the force is attractive.

The size of the force does not care which charge is which. A negative and a positive charge at 8 cm feel exactly the same strength of pull as a positive and a negative at 8 cm, and the pull on each object is equal and opposite, following Newton's third law. Swapping the two charge inputs in the calculator leaves the force magnitude unchanged.

Everyday static electricity leans on this rule. A balloon rubbed on hair picks up extra electrons and goes negative, while the hair loses them and goes positive, which is why the hair stands up toward the balloon. A neutral wall still attracts the balloon through a rearrangement of its surface charges, but that induced-charge effect is a step beyond the two-charge law on this page.

Coulomb force against gravity

It is natural to ask whether gravity matters alongside electricity. OpenStax College Physics 2e runs the comparison for the electron and proton in a hydrogen atom at their 0.530 x 10^-10 m separation. The electric force comes out to 8.19 x 10^-8 N. The gravitational force, from Newton's law with G = 6.67 x 10^-11, comes out to about 3.61 x 10^-47 N.

Divide the two and the electric force wins by a factor of roughly 2.3 x 10^39. Written out, that is a 2 followed by 39 digits. Gravity only wins when objects carry enormous, balanced amounts of positive and negative charge, which is why planets follow gravity while atoms follow electricity. Ordinary objects stay near neutral, so the huge internal electric forces cancel out and leave gravity in charge of the big picture.

That cancellation also explains why static shocks feel dramatic anyway. A doorknob spark moves only a fraction of a microcoulomb, but it moves it across a gap of a few millimeters, where the inverse square term is enormous. Keep the distance small and even tiny charges pack a punch you can feel in your fingertip.

What the calculator returns

The defaults match a pair of small charged beads: 50 nC and 150 nC sitting 2 cm apart. Pick the unit next to each box, type the values, and the results area fills in as you go. Here is what the default case gives.

Three habits keep the answers honest. Convert the distance to meters or pick the matching unit from the dropdown, since centimeters entered as meters throw the result off by 10,000. Check the sign display, because a repel result where you expected attract usually means a minus sign went missing. And sanity check the scale: forces above a newton from handheld charges mean the charges are large or the gap is small, so re-read the inputs before trusting the number.