Electric Force Equation: Coulomb’s Law, 7 Formulas, and Examples
The electric force equation is F = kq₁q₂ / r², known as Coulomb’s law. It gives the force in newtons (N) between 2 point charges in coulombs (C) separated by a distance r in meters (m). The 7 formulas below cover charges, permittivity, vectors, electric fields, voltage, media, and multiple charges.
For a reliable reference, see OpenStax presents Coulomb law for the force between point charges.
Separated charges can also store energy; see Electric Potential Energy Formula: Equations, Units, Sign Rules, and Worked Examples.
What Is the Electric Force Equation and What Are Examples of It?
The electric force equation is the formula that calculates the force between 2 electric charges from the size of the charges and the distance between them. Charles-Augustin de Coulomb published the relationship in 1785, so it is called Coulomb’s law. According to OpenStax University Physics Volume 2, the force between 2 point charges is proportional to the product of the charges and inversely proportional to the square of the distance.
Electric force is a vector quantity, so it has a magnitude and a direction. The force acts along the straight line that joins the 2 charges. Opposite charges attract, and like charges repel.
Doubling the distance reduces the force to one quarter. Doubling 1 charge doubles the force.
Examples include the force between an electron and a proton in a hydrogen atom, the force between 2 charged balloons, and the force between 2 protons in a nucleus.
The table below defines each symbol in the magnitude form F = kq₁q₂ / r² and lists its SI unit. In this form, q₁ and q₂ are the absolute values of the charges.
| Symbol | Meaning | SI Unit |
|---|---|---|
| F | Magnitude of the electric force | newton (N) |
| k | Coulomb constant | N·m²/C² |
| q₁ | Charge magnitude of the first object | coulomb (C) |
| q₂ | Charge magnitude of the second object | coulomb (C) |
| r | Distance between the centers of the charges | meter (m) |
What Are the 7 Electric Force Formulas?
The 7 electric force formulas use charges and distance, permittivity, vectors, electric field, voltage, medium, or multiple charges. The table below lists the equation, known quantities, and condition for each formula.
Symbols in the table: ε₀ is the vacuum permittivity (F/m), ε_r is the relative permittivity (no unit), E is the electric field (N/C), V is the potential difference (V), d is the plate separation (m), and r̂ is the unit vector from the source charge to the charge that feels the force.
| Formula | Equation | Known Quantities | Applies When |
|---|---|---|---|
| Coulomb’s law | F = kq₁q₂ / r² | q₁, q₂, r | 2 point charges are in a vacuum or air |
| Permittivity form | F = q₁q₂ / (4π ε₀ r²) | q₁, q₂, r, ε₀ | The equation uses the SI constant ε₀ |
| Vector form | F⃗ = k q₁q₂ r̂ / r² | q₁, q₂, r, direction | The direction of the force is required |
| Field form | F = qE | q, E | The electric field at the charge location is known |
| Parallel plate form | F = qV / d | q, V, d | A charge sits in the uniform field between 2 parallel plates |
| Medium form | F = kq₁q₂ / (ε_r r²) | q₁, q₂, r, ε_r | The charges sit in a dielectric material |
| Superposition | F⃗_net = F⃗₁ + F⃗₂ + … + F⃗_n | Individual force vectors | 3 or more charges interact |
How Do You Use Coulomb’s Law to Find Electric Force?
Electric force is found by multiplying the Coulomb constant by the product of the 2 charge magnitudes and dividing by the squared distance.
- Convert the charges to coulombs.
- Convert the distance to meters.
- Multiply the magnitudes of the 2 charges.
- Multiply the product by k = 8.99 × 10⁹ N·m²/C².
- Divide the result by r².
- Assign the direction, with attraction for opposite signs and repulsion for like signs.
Example: Two charges of +3 μC and +5 μC sit 0.20 m apart. The force is 8.99 × 10⁹ × (3 × 10⁻⁶)(5 × 10⁻⁶) / 0.20² = 3.37 N. The force is repulsive, because both charges are positive.
Moving the same charges to 0.40 m reduces the force to 3.37 / 4 ≈ 0.84 N.
What Is the Value of the Constant k in the Electric Force Equation?
The Coulomb constant is k ≈ 8.988 × 10⁹ N·m²/C², and it equals 1 / (4π ε₀). NIST publishes the CODATA recommended value of k as approximately 8.98755 × 10⁹ N·m²/C². The vacuum permittivity ε₀ is approximately 8.854 × 10⁻¹² F/m.
The table below lists 3 rounding levels of k and the typical use of each.
| Rounding Level | Value (N·m²/C²) | Typical Use |
|---|---|---|
| 4 significant figures | 8.988 × 10⁹ | Research and precise calculations |
| 3 significant figures | 8.99 × 10⁹ | Textbook problems |
| 2 significant figures | 9.0 × 10⁹ | Estimates and quick checks |
The elementary charge is e = 1.602176634 × 10⁻¹⁹ C. The 2019 SI redefinition fixed this value as exact.
What Is the Vector Form of the Electric Force Equation?
The vector form is F⃗ = k q₁q₂ r̂ / r², where r̂ is the unit vector pointing from q₂ to q₁. The equation gives the force on q₁ caused by q₂. The charges carry their signs in this form.
A positive product q₁q₂ gives a force along r̂, which is repulsion. A negative product gives a force opposite to r̂, which is attraction.
The equivalent form is F⃗ = k q₁q₂ r⃗ / r³, where r⃗ is the displacement vector from q₂ to q₁. The force on q₂ caused by q₁ equals the negative of the force on q₁, as Newton’s third law states.
How Do You Find the Direction of the Electric Force?
The direction of the electric force is along the line joining the 2 charges, away from the other charge for like signs and toward it for opposite signs. The table below lists the 4 sign combinations, the product of the charges, the type of force, and the direction of the force on q₁.
| q₁ | q₂ | Product q₁q₂ | Type of Force | Direction of Force on q₁ |
|---|---|---|---|---|
| Positive | Positive | Positive | Repulsion | Away from q₂ |
| Negative | Negative | Positive | Repulsion | Away from q₂ |
| Positive | Negative | Negative | Attraction | Toward q₂ |
| Negative | Positive | Negative | Attraction | Toward q₂ |
How Do You Find Electric Force From an Electric Field?
Electric force is found by multiplying the charge by the electric field at its location (F = qE). The unit of electric field is newtons per coulomb (N/C), which equals volts per meter (V/m).
The force on a positive charge points in the direction of the field. The force on a negative charge points opposite to the field. The field of a point charge Q is E = kQ / r², so F = qE reproduces Coulomb’s law.
Example: An electron (q = −1.602 × 10⁻¹⁹ C) sits in a field of 1,000 N/C pointing east. The force magnitude is 1.602 × 10⁻¹⁶ N, directed west. The acceleration is 1.602 × 10⁻¹⁶ / 9.109 × 10⁻³¹ ≈ 1.76 × 10¹⁴ m/s².
What Are the Electric Field Formulas for Common Charge Distributions?
Common charge distributions have 6 standard electric field formulas, and each gives the force through F = qE. The table below lists the field magnitude and condition for each distribution. Q is the total charge (C), λ is the charge per unit length (C/m), σ is the charge per unit area (C/m²), and r is the distance from the source (m).
| Distribution | Field Magnitude E | Condition |
|---|---|---|
| Point charge | E = kQ / r² | Any distance r from the point |
| Uniformly charged sphere or shell, outside | E = kQ / r² | r is greater than the radius |
| Uniformly charged shell, inside | E = 0 | r is less than the radius |
| Infinite line of charge | E = λ / (2π ε₀ r) | r is much smaller than the line length |
| Infinite insulating sheet of charge | E = σ / (2 ε₀) | The distance is small relative to the sheet size |
| Surface of a conductor | E = σ / ε₀ | The point lies just outside the surface |
How Do You Find Electric Force Between Parallel Plates?
Electric force between parallel plates is found with F = qE = qV / d, where V is the potential difference and d is the plate separation. The field between the plates is uniform, away from the edges.
Example: A 2 nC charge sits between plates 0.02 m apart with a potential difference of 200 V. The field is 200 / 0.02 = 10,000 V/m, which equals 10,000 N/C. The force is 2 × 10⁻⁹ × 10,000 = 2 × 10⁻⁵ N.
How Do You Find Net Electric Force From Multiple Charges?
Net electric force is found by adding the individual force vectors from each charge, known as the superposition principle. Each force is calculated with Coulomb’s law for 1 pair of charges. Forces along 1 line add with signs, and forces at angles add by components.
Example in 1 dimension: The charges q₁ = +4 μC at x = 0, q₂ = +1 μC at x = 0.10 m, and q₃ = −2 μC at x = 0.30 m lie on the x-axis. The net force on q₂ follows 3 steps:
- Force from q₁: 8.99 × 10⁹ × (4 × 10⁻⁶)(1 × 10⁻⁶) / 0.10² = 3.60 N toward +x, as repulsion.
- Force from q₃: 8.99 × 10⁹ × (1 × 10⁻⁶)(2 × 10⁻⁶) / 0.20² = 0.45 N toward +x, as attraction.
- Net force: 3.60 + 0.45 = 4.05 N toward +x.
Example in 2 dimensions: A +1 μC charge sits at the origin. A +3 μC charge sits at (0.30 m, 0), and a +4 μC charge sits at (0, 0.40 m).
- Force from the +3 μC charge: 8.99 × 10⁹ × (3 × 10⁻⁶)(1 × 10⁻⁶) / 0.30² = 0.30 N in the −x direction.
- Force from the +4 μC charge: 8.99 × 10⁹ × (4 × 10⁻⁶)(1 × 10⁻⁶) / 0.40² = 0.225 N in the −y direction.
- Net magnitude: √(0.30² + 0.225²) = 0.375 N.
- Direction: arctan(0.225 / 0.30) = 36.9° below the negative x-axis.
Where Is the Net Electric Force Zero Between 2 Charges?
The net force on a third charge is zero at the point where the 2 forces have equal magnitudes and opposite directions. For 2 like charges separated by a distance L, the point lies between them, closer to the smaller charge. The distance from q₁ is x = L√q₁ / (√q₁ + √q₂).
Example: Charges of 4 μC and 1 μC sit 0.30 m apart. The zero-force point is 0.30 × 2 / (2 + 1) = 0.20 m from the 4 μC charge.
For 2 opposite charges, the zero-force point lies outside the pair, on the side of the charge with the smaller magnitude.
How Does a Medium Change the Electric Force?
A dielectric medium reduces the electric force by the factor of its relative permittivity ε_r (F = kq₁q₂ / (ε_r r²)). The table below lists approximate relative permittivity at room temperature and the force ratio compared with a vacuum, for the same charges and distance.
| Medium | Relative Permittivity ε_r | Force Compared With Vacuum |
|---|---|---|
| Vacuum | 1 | 1 |
| Dry air | 1.0006 | 0.9994 |
| Polyethylene | 2.3 | 0.43 |
| Water at 20 °C | 80 | 0.0125 |
Example: The 3.37 N force between the +3 μC and +5 μC charges at 0.20 m in a vacuum falls to 3.37 / 80 ≈ 0.042 N in water. This reduction explains why ionic compounds such as sodium chloride dissolve in water.
What Is the Difference Between Electric Force and Electric Field?
Electric force is the interaction that acts on a specific charge, and electric field is the force per unit charge at a location in space. The table below compares the 2 quantities by 5 features.
| Feature | Electric Force | Electric Field |
|---|---|---|
| Symbol | F | E |
| Equation for a point charge | F = kq₁q₂ / r² | E = kQ / r² |
| SI unit | newton (N) | newton per coulomb (N/C) |
| Depends on the charge that feels it | Yes | No |
| Quantity type | Vector | Vector |
The relationship between the 2 quantities is F = qE.
How Do the Electric Force and Gravitational Force Equations Compare?
Both equations follow an inverse-square law, but the electric force between a proton and an electron is about 2.27 × 10³⁹ times stronger than the gravitational force. The distance cancels in the ratio kе² / (G mₚ mₑ). The table below compares the 2 equations by 6 features.
| Feature | Electric Force (Coulomb’s Law) | Gravitational Force (Newton’s Law) |
|---|---|---|
| Equation | F = kq₁q₂ / r² | F = G m₁m₂ / r² |
| Constant | k ≈ 8.988 × 10⁹ N·m²/C² | G ≈ 6.674 × 10⁻¹¹ N·m²/kg² |
| Source property | Charge (C) | Mass (kg) |
| Direction | Attractive or repulsive | Attractive only |
| Screening by conductors | Possible | Not possible |
| Dominant scale | Atoms and molecules | Planets, stars, and galaxies |
What Is the Electric Force on a Moving Charge?
The total electromagnetic force on a moving charge is F⃗ = q(E⃗ + v⃗ × B⃗), known as the Lorentz force. The term qE⃗ is the electric force. The term q v⃗ × B⃗ is the magnetic force, and it equals 0 for a charge at rest.
Coulomb’s law applies exactly to stationary charges. Moving charges require the Lorentz force and relativistic corrections.
How Do Electric Force and Electric Potential Energy Relate?
Electric potential energy is U = kq₁q₂ / r, and the electric force equals the negative derivative of U with respect to r (F = −dU/dr). The unit of U is the joule (J), with U = 0 at infinite separation. Like charges have positive U, and opposite charges have negative U.
Example: The +3 μC and +5 μC charges at 0.20 m have U = 8.99 × 10⁹ × (3 × 10⁻⁶)(5 × 10⁻⁶) / 0.20 ≈ 0.67 J. The force is U / r = 0.67 / 0.20 ≈ 3.37 N, which matches the Coulomb’s law result.
How Is the Electric Force Equation Derived and Tested?
Coulomb measured the electric force with a torsion balance and published the inverse-square law in 1785. A torsion balance measures force from the twist angle of a thin suspension fiber. Henry Cavendish measured the same inverse-square dependence earlier, but his results were published in 1879.
Gauss’s law gives the same result. For a point charge Q, the field on a sphere of radius r satisfies E × 4π r² = Q / ε₀. The solution is E = Q / (4π ε₀ r²), and F = qE reproduces Coulomb’s law.
Experiments confirmed the exponent 2 to high precision:
- Plimpton and Lawton reported in 1936 that the exponent equals 2 within 2 parts in 10⁹.
- Williams, Faller, and Hill tightened the limit in 1971 to about 1 part in 10¹⁵.
What Are the Units of the Electric Force Equation?
The SI unit of electric force is the newton (N), which equals 1 kg·m/s². The table below lists the quantity, symbol, SI unit, and conversion for 6 terms in the electric force equations.
| Quantity | Symbol | SI Unit | Conversion or Value |
|---|---|---|---|
| Electric force | F | newton (N) | 1 N = 1 kg·m/s² |
| Charge | q | coulomb (C) | 1 C = 1 A·s, or about 6.242 × 10¹⁸ elementary charges |
| Distance | r | meter (m) | 1 cm = 0.01 m |
| Coulomb constant | k | N·m²/C² | About 8.988 × 10⁹ |
| Electric field | E | N/C | 1 N/C = 1 V/m |
| Vacuum permittivity | ε₀ | F/m | About 8.854 × 10⁻¹² |
Charge prefixes include 1 μC = 10⁻⁶ C, 1 nC = 10⁻⁹ C, and 1 pC = 10⁻¹² C.
What Are Examples of Electric Force Calculations?
Electric force calculations range from 10⁻⁸ N at atomic scale to 10⁹ N for 2 charges of 1 C. The table below lists 5 situations with the charges, distance, force, and type of force. Values use k = 8.99 × 10⁹ N·m²/C².
| Situation | q₁ | q₂ | Distance r | Force | Type of Force |
|---|---|---|---|---|---|
| Electron and proton in hydrogen | −1.602 × 10⁻¹⁹ C | +1.602 × 10⁻¹⁹ C | 5.29 × 10⁻¹¹ m | 8.2 × 10⁻⁸ N | Attraction |
| 2 protons in a nucleus | +1.602 × 10⁻¹⁹ C | +1.602 × 10⁻¹⁹ C | 1.0 × 10⁻¹⁵ m | 231 N | Repulsion |
| 2 charged balloons | +50 nC | +50 nC | 0.10 m | 2.25 × 10⁻³ N | Repulsion |
| 2 small charged spheres | +1 μC | −1 μC | 1.0 m | 8.99 × 10⁻³ N | Attraction |
| 2 large charges | +1 C | +1 C | 1.0 m | 8.99 × 10⁹ N | Repulsion |
The force of 231 N between 2 protons at 1 femtometer equals the weight of a mass of about 24 kg near Earth’s surface. The last row shows that 1 coulomb is a large charge.
What Are Real-Life Applications of the Electric Force Equation?
The electric force equation applies to 7 technologies and natural phenomena. Examples include:
- Electrostatic precipitators, which remove particles from industrial exhaust.
- Laser printers and photocopiers, which attract toner to charged areas of a drum.
- Electrostatic spray painting, which attracts charged paint droplets to a grounded surface.
- Inkjet printers, which deflect charged ink droplets with electric fields.
- Particle accelerators, which accelerate charged particles with electric fields.
- Atoms and molecules, where electric attraction between nuclei and electrons forms atomic structure and chemical bonds.
- Lightning, which occurs when the electric field exceeds about 3 × 10⁶ V/m, the dielectric strength of dry air.
How Do You Measure Electric Force Experimentally?
Electric force is measured by recording the deflection of a charged object and applying a force balance or a torsion calculation. Instruments include torsion balances, electronic force sensors, and suspended charged balls.
A hanging charged ball deflects by an angle θ from the vertical when another charge acts horizontally on it. The electric force is F = mg tan θ.
- Hang a charged ball from an insulating thread.
- Bring a second charged object to a measured distance r.
- Measure the deflection angle θ from the vertical.
- Calculate the force with F = mg tan θ.
- Repeat at 3 distances to test the 1 / r² relationship.
Example: A 1.0 g ball deflects by 10°. The force is 1.0 × 10⁻³ × 9.8 × tan 10° ≈ 1.7 × 10⁻³ N.
What Are the Limits of the Electric Force Equation?
The electric force equation applies exactly to stationary point charges and to spherically symmetric charge distributions. Conditions include:
- Point charges: the object sizes are much smaller than the distance r.
- Spherical distributions: a uniformly charged sphere acts as a point charge at its center for points outside the sphere.
- Close conductors: charges redistribute on conducting spheres, so the formula is approximate at small separations.
- Moving charges: the Lorentz force and relativistic corrections apply.
- Very short distances: quantum electrodynamics adds corrections.
- Dielectric media: the factor ε_r applies.
What Are the Common Mistakes When Using the Electric Force Equation?
6 common mistakes cause incorrect electric force results. Rainson, Tranströmer, and Viennot published a study in the American Journal of Physics in 1994 and found that students have difficulty combining electric fields from more than 1 source.
The mistakes include:
- Forgetting to convert μC, nC, and cm to C and m. Example: 5 cm equals 0.05 m.
- Using r instead of r² in the denominator.
- Substituting signed charges into the magnitude formula, which produces a negative magnitude.
- Adding force magnitudes instead of force vectors.
- Measuring the distance to the surface of a sphere instead of its center.
- Applying the vacuum formula to charges in a dielectric medium.
What Do People Also Ask About the Electric Force Equation?
What is the electric force equation?
The electric force equation is F = kq₁q₂ / r², which gives the force between 2 point charges.
What is the formula for electric force with an electric field?
The formula is F = qE, where q is the charge and E is the electric field.
What is the electric force equation with voltage?
The equation is F = qV / d for a charge between 2 parallel plates with a potential difference V and a separation d.
What is the unit of electric force?
The SI unit of electric force is the newton (N).
What is the value of k in the electric force equation?
The Coulomb constant is approximately 8.988 × 10⁹ N·m²/C².
Is electric force a vector or a scalar?
Electric force is a vector, and it acts along the line joining the charges.
What does electric force depend on?
Electric force depends on the magnitudes of the charges, the distance between them, and the medium.
What happens to electric force when the distance doubles?
The electric force falls to one quarter of its original value.
Is Coulomb’s law the same as the electric force equation?
Coulomb’s law is the electric force equation for 2 point charges.
Can electric force be negative?
Electric force carries a negative sign in the vector form when the force is attractive.
What is the electric force between an electron and a proton in hydrogen?
The force is about 8.2 × 10⁻⁸ N at the Bohr radius of 5.29 × 10⁻¹¹ m.
How strong is electric force compared with gravity?
The electric force between a proton and an electron is about 2.27 × 10³⁹ times stronger than the gravitational force.
Who discovered the electric force equation?
Charles-Augustin de Coulomb published the equation in 1785.
What Is the Electric Force Equation and How Is It Applied?
The electric force equation is F = kq₁q₂ / r², which gives the force between 2 point charges in newtons. The 7 formulas use charges and distance, permittivity, vectors, electric field, voltage, medium, or multiple charges. Like charges repel, and opposite charges attract. Doubling the distance reduces the force to one quarter. The constant k is approximately 8.988 × 10⁹ N·m²/C², and a dielectric medium divides the force by ε_r. Applications include electrostatic precipitators, laser printers, inkjet printers, particle accelerators, and atomic bonding.
Title: Newton’s Third Law of Motion Examples: Definition, 20 Examples, and Force Pairs
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