Coulomb's Law Calculator
Calculate electric forces, fields, and potential energy between charged particles with vector analysis and superposition.
Calculate electric forces, fields, and potential energy between charged particles with vector analysis and superposition.
From the static shock you feel touching a doorknob to the fundamental forces holding atoms together, electrostatic interactions govern countless phenomena in physics and everyday life. Understanding Coulomb's Law—the mathematical relationship describing how electrically charged objects attract or repel each other—is essential for anyone studying physics, chemistry, electrical engineering, or simply curious about how the world works at the atomic and molecular scale. The fundamental relationship is F = k·|q₁q₂|/r², where F is the electrostatic force (newtons), k ≈ 8.99×10⁹ N·m²/C² is Coulomb's constant, q₁ and q₂ are the charges (coulombs), and r is the distance between them (meters). Like charges (both + or both −) repel each other, while opposite charges (+ and −) attract. Understanding Coulomb's Law helps you calculate electric forces, understand electrostatic interactions, and work with charged particles. This tool solves Coulomb's Law problems—you provide charges and distance, and it calculates force, electric field, potential energy, and superposition effects, showing step-by-step solutions and helping you verify your work.
For students and researchers, this tool demonstrates practical applications of Coulomb's Law, electric forces, and electrostatic principles. The Coulomb's Law calculations show how force relates to charges and distance (F = k·|q₁q₂|/r²), how electric field relates to charge and distance (E = kq/r²), how potential energy relates to charges and distance (U = kq₁q₂/r), how superposition works for multiple charges (F_total = Σ Fᵢ), and how the inverse-square law affects force (doubling distance reduces force by 4×). Students can use this tool to verify homework calculations, understand how electrostatic forces work, explore concepts like attractive vs repulsive forces, and see how different parameters affect force. Researchers can apply electrostatic principles to analyze charge distributions, predict forces, and understand electrostatic phenomena. The visualization helps students and researchers see how charges interact.
For engineers and practitioners, Coulomb's Law provides essential tools for analyzing electrical systems, designing devices, and understanding electrostatic behavior in real-world applications. Electrical engineers use Coulomb's Law to design capacitors, analyze electric fields, and understand electrostatic phenomena. Electronics engineers use Coulomb's Law to understand charge interactions, design circuits, and analyze device behavior. These applications require understanding how to apply Coulomb's Law formulas, interpret results, and account for real-world factors like medium permittivity, charge distributions, and geometry effects. However, for engineering applications, consider additional factors and safety margins beyond simple point-charge calculations.
For the common person, this tool answers practical electrostatic questions: How strong is the force between two charges? Why do opposite charges attract? The tool solves Coulomb's Law problems using electrostatic formulas, showing how these parameters affect force. Taxpayers and budget-conscious individuals can use electrostatic principles to understand electrical phenomena, assess electrical safety, and make informed decisions about electrical equipment. These concepts help you understand how electricity works and how to solve electrostatic problems, fundamental skills in understanding physics and chemistry.
This calculator is for educational purposes—learning and practice with Coulomb's Law formulas. For engineering applications, consider additional factors like idealized point-charge conditions (no extended objects, complex geometries, or material effects), not an electrical design or safety compliance tool (does not account for real-world factors, safety margins, or regulatory requirements), and real electrical design requires professional analysis. This tool assumes ideal point-charge conditions (no extended objects, no complex geometries, no material effects)—simplifications that may not apply to real-world scenarios. Always verify important results independently and consult engineering standards for design applications. Real electrical design requires professional analysis and appropriate safety considerations.
Coulomb's Law states that the electrostatic force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them: F = k·|q₁q₂|/r². Like charges (both + or both −) repel each other, opposite charges (+ and −) attract. The constant k ≈ 8.99×10⁹ N·m²/C² in vacuum. This means larger charges create stronger forces, and force decreases rapidly as distance increases. Understanding Coulomb's Law helps you work with electrostatic forces and solve electrostatic problems.
Electric charge is a fundamental property of matter, measured in coulombs (C). Charges come in two types: positive (+) and negative (−). Protons carry positive charge (+1.6×10⁻¹⁹ C), electrons carry negative charge (−1.6×10⁻¹⁹ C). Like charges (both + or both −) repel each other, while opposite charges (+ and −) attract. In calculations, the sign of the charge affects whether the force is attractive or repulsive. Common units: 1 microcoulomb (µC) = 10⁻⁶ C, 1 nanocoulomb (nC) = 10⁻⁹ C. Understanding charge helps you understand how electrostatic forces work.
The separation distance (r) between charges is measured in meters (m). Coulomb's Law assumes point charges—charged objects so small that their size is negligible compared to the distance between them. For macroscopic objects, r is measured center-to-center. The force follows an inverse-square law: doubling the distance reduces the force by a factor of four (2² = 4), tripling the distance reduces force by nine (3² = 9), and so on. This inverse-square behavior is fundamental to both electrostatics and gravity. Understanding distance effects helps you predict how forces change.
Vacuum permittivity ε₀ is a constant that appears in Coulomb's Law. Relative permittivity εᵣ describes how much a medium (for example, air, glass, water) reduces the effective electric field and force compared to vacuum. The effective permittivity is ε = εᵣ × ε₀. The calculator uses the appropriate value of k or ε based on medium or εᵣ. For example, air has εᵣ ≈ 1.0006 (essentially same as vacuum), while water has εᵣ ≈ 80, making forces in water about 80× weaker than in vacuum. Understanding permittivity helps you understand how materials affect electrostatic forces.
The sign of q₁ × q₂ determines the interaction: positive product → like charges → repulsive force. Negative product → opposite charges → attractive force. The tool can label the interaction as "Attractive" or "Repulsive" and show the force magnitude separately. Unlike gravity (which is always attractive), electrostatic forces can be either attractive or repulsive, making them fundamentally different from gravitational interactions. Understanding attraction vs repulsion helps you interpret force directions correctly.
Electric field E at distance r from charge q is E = kq/r². The force on a test charge q_test is F = q_test × E = q_test × kq/r². This shows Coulomb's Law (F = kq₁q₂/r²) is equivalent to force = charge × field. Electric field is measured in N/C or equivalently V/m. Fields allow us to analyze forces systematically without tracking every charge pair. Understanding electric field helps you understand how charges create forces at a distance.
Electric potential energy U = kq₁q₂/r is the work needed to bring two charges from infinite separation to distance r. It's positive for like charges (repulsive, work required), negative for opposite charges (attractive, energy released). The relationship to force is F = −dU/dr (force is negative gradient of potential energy). As charges move closer together, potential energy of like charges increases (energy stored), opposite charges decreases (energy released). Understanding potential energy helps you understand energy in electrostatic systems.
For a test charge q₀ influenced by multiple charges qᵢ: F_total = Σ Fᵢ. Each Fᵢ is computed using Coulomb's Law for the pair (q₀, qᵢ). The tool adds these forces vectorially to show the net effect, allowing analysis of complex charge distributions using the superposition principle. This is essential for understanding electric fields from charge distributions and more advanced topics like Gauss's Law. Understanding superposition helps you analyze complex charge systems.
The inverse-square law (F ∝ 1/r²) arises from geometry: electric field lines spread out uniformly in all directions from a point charge. As you move farther away, the same 'flux' spreads over a sphere of radius r, which has surface area 4πr². The field strength (and hence force) at distance r is diluted by the factor 1/r². Doubling distance spreads the field over 4× the area, reducing force by 4×. This inverse-square behavior is fundamental to both electrostatics and gravity. Understanding this geometric explanation helps you understand why forces decrease with distance.
Select the calculation mode: Electric Force (F = k·|q₁q₂|/r²), Solve for Unknown (q₁, q₂, r, or F), Electric Field at a Point, Superposition (Multiple Charges), Potential Energy (U = k·q₁q₂/r), or Force Components (Fₓ, Fᵧ). Each mode focuses on different aspects of Coulomb's Law. Choose the mode that matches your problem.
For electric force scenarios, enter charge q₁ (magnitude and sign in coulombs), charge q₂ (magnitude and sign in coulombs), and distance r between them (in meters). Ensure you use correct units: charge in coulombs (C—convert µC by multiplying by 10⁻⁶, nC by 10⁻⁹), distance in meters (m—convert cm by dividing by 100, mm by dividing by 1000). Unit conversion errors are the #1 source of incorrect results.
Choose medium preset: Vacuum/Air (εᵣ ≈ 1) or Custom (enter your own relative permittivity εᵣ). For vacuum/air, the tool automatically uses k ≈ 8.99×10⁹ N·m²/C². For custom, you can override relative permittivity (εᵣ). The effective Coulomb constant becomes k_eff = k/εᵣ. Medium affects force strength—forces in water (εᵣ ≈ 80) are about 80× weaker than in vacuum.
For vector/coordinate mode, enter positions of each charge (x, y coordinates in meters). The tool calculates force components (Fₓ, Fᵧ) and overall magnitude. This is essential for understanding how multiple charges create net forces in 2D and 3D space. Use this mode to practice breaking forces into components.
For superposition scenarios, enter the test charge (q₀) and a list of other charges (q₁, q₂, q₃, …) and their positions. The tool calculates the net force on the test charge from all other charges, with optional breakdown by contribution from each charge. This allows analysis of complex charge distributions using the superposition principle.
For solve unknown scenarios, select which variable to solve for (q₁, q₂, r, or F) and enter the other three known values. The tool calculates the unknown using algebraic rearrangements of Coulomb's Law. This helps you solve problems where one parameter is unknown.
Optionally set the number of decimal places for results (2, 3, 4, or 6). This controls precision of displayed values. For most applications, 2–3 decimal places are sufficient. Higher precision (4–6 decimals) is useful for precision calculations or academic work.
Click "Calculate" or submit the form to solve the Coulomb's Law equations. The tool displays: (1) Force magnitude and direction (attractive or repulsive), (2) Electric field (if field mode), (3) Potential energy (if energy mode), (4) Force components (if vector mode), (5) Formula used—which equation was applied, (6) Step-by-step calculation—algebraic steps showing how values were calculated, (7) Visualization—charge positions and force vectors (if coordinate mode). Review the results to understand electrostatic interactions and verify that values make physical sense.
The key formulas for Coulomb's Law calculations:
Coulomb's Law (Magnitude): F = k · |q₁ × q₂| / r²
Electrostatic force between two point charges (k = 8.99×10⁹ N·m²/C² in vacuum)
Using Permittivity: F = (1 / (4π ε)) · |q₁ q₂| / r²
Alternative form using permittivity (ε = εᵣ × ε₀)
Electric Field: E = kq / r²
Electric field created by charge q at distance r (N/C or V/m)
Potential Energy: U = kq₁q₂ / r
Electric potential energy stored in a system of two charges (joules)
Coulomb's Constant (Vacuum): k = 8.99×10⁹ N·m²/C²
Fundamental constant relating charge to force
Effective Constant (Medium): k_eff = k / εᵣ
Coulomb's constant adjusted for medium permittivity
These formulas are interconnected—the solver uses algebraic relationships to convert between force, electric field, potential energy, and charge parameters. Understanding which formula to use helps you solve problems manually and interpret solver results.
The solver uses different strategies depending on the calculation mode:
Electric Force Mode:
Calculate F = k·|q₁q₂|/r² using effective k based on medium
Determine direction: attractive if q₁q₂ < 0, repulsive if q₁q₂ > 0
If coordinates provided: calculate force components Fₓ, Fᵧ using vector decomposition
Electric Field Mode:
Calculate E = kq/r² for field created by charge q at evaluation point
If multiple charges: use superposition E_total = Σ Eᵢ
Calculate field components Eₓ, Eᵧ if coordinates provided
Potential Energy Mode:
Calculate U = kq₁q₂/r for energy stored in charge system
Determine sign: positive for like charges, negative for opposite charges
Superposition Mode:
Calculate force from each charge separately: Fᵢ = k·|q₀qᵢ|/rᵢ²
Add forces vectorially: F_total = Σ Fᵢ
Calculate net force components: Fₓ_total = Σ Fₓᵢ, Fᵧ_total = Σ Fᵧᵢ
The solver uses this strategy to calculate electrostatic parameters. Understanding this helps you interpret results and predict electrostatic behavior.
Let's calculate the force between two charges:
Given: q₁ = +2.0 µC = 2×10⁻⁶ C, q₂ = -3.0 µC = -3×10⁻⁶ C, r = 0.5 m, Medium: air (εᵣ ≈ 1)
Find: Electrostatic force F
Step 1: Calculate charge product
|q₁q₂| = |(2×10⁻⁶)(−3×10⁻⁶)| = 6×10⁻¹² C²
Step 2: Calculate force using Coulomb's Law
F = k·|q₁q₂|/r² = (8.99×10⁹ N·m²/C²) × (6×10⁻¹² C²) / (0.5 m)²
F = (8.99×10⁹) × (6×10⁻¹²) / 0.25 = 0.216 N
Result:
The force is 0.216 N (attractive) because charges have opposite signs (+ and −). This demonstrates how Coulomb's Law calculates force from charges and distance.
This example demonstrates how to calculate electrostatic force using Coulomb's Law. The charge product is calculated first, then the force formula is applied. Understanding this helps you solve basic Coulomb's Law problems.
Let's see how doubling distance affects force:
Given: Same charges as above (q₁ = +2.0 µC, q₂ = -3.0 µC), but r doubled to 1.0 m
Find: New force F
Step 1: Calculate force with new distance
F = k·|q₁q₂|/r² = (8.99×10⁹) × (6×10⁻¹²) / (1.0 m)² = 0.054 N
Step 2: Compare to original force
Original force: 0.216 N, New force: 0.054 N
Ratio: 0.216 / 0.054 = 4 (force reduced by 4×)
Result:
Doubling distance reduces force by a factor of 4 (2² = 4), demonstrating the inverse-square relationship. This shows why electrostatic forces are strong at short range but negligible at large distances.
This example demonstrates the inverse-square law. When distance doubles, force decreases by 4× (not 2×). Understanding this helps you predict how forces change with distance.
Let's calculate the electric field created by a charge:
Given: Charge q = +5.0 µC = 5×10⁻⁶ C, Distance r = 0.3 m
Find: Electric field E
Step 1: Calculate electric field
E = kq/r² = (8.99×10⁹ N·m²/C²) × (5×10⁻⁶ C) / (0.3 m)²
E = (8.99×10⁹) × (5×10⁻⁶) / 0.09 = 4.99×10⁵ N/C
Result:
The electric field is 4.99×10⁵ N/C (or 4.99×10⁵ V/m) pointing away from the positive charge. A test charge placed at this point would experience a force F = q_test × E.
This example demonstrates how to calculate electric field. The field is force per unit charge, so E = kq/r². Understanding this helps you connect force and field concepts.
A student needs to solve: "Two charges of +3 µC and -5 µC are separated by 0.2 meters. What is the force between them?" Using the tool with electric force mode, entering q₁ = 3×10⁻⁶ C, q₂ = -5×10⁻⁶ C, and r = 0.2 m, the tool calculates F = 3.37 N (attractive). The student learns that opposite charges attract, and can see how different charges and distances correspond to different forces. This helps them understand how Coulomb's Law works and how to solve electrostatic problems.
A physics student explores: "How does force change when distance doubles?" Using the tool with electric force mode, comparing r = 0.1 m vs r = 0.2 m (same charges), they can see that doubling distance reduces force by 4× (not 2×). The student learns that the inverse-square law means F ∝ 1/r², helping them understand why electrostatic forces are strong at short range but negligible at large distances.
An engineer needs to analyze: "What's the electric field at a point from three charges?" Using the tool with superposition mode, entering multiple charges and their positions, the tool calculates net electric field E_total = Σ Eᵢ. The engineer learns how superposition works and can see how different charge configurations affect field strength. Note: This is for educational purposes—real engineering requires additional factors and professional analysis.
A person wants to understand: "Why do opposite charges attract?" Using the tool with electric force mode, comparing like charges (both +) vs opposite charges (+ and -), they can see that like charges produce repulsive forces while opposite charges produce attractive forces. The person learns that charge signs determine force direction, helping them understand basic electrostatic interactions.
A researcher analyzes: "How strong is the force holding Na⁺ and Cl⁻ ions together in NaCl?" Using the tool with electric force mode, entering q₁ = +1.6×10⁻¹⁹ C (Na⁺), q₂ = -1.6×10⁻¹⁹ C (Cl⁻), and r = 2.8×10⁻¹⁰ m, they can see that F ≈ 2.9×10⁻⁹ N. While tiny by everyday standards, this force is enormous at the atomic scale, explaining why ionic compounds have high melting points. This helps understand chemical bonding.
A student explores: "How does water affect electrostatic forces?" Using the tool with electric force mode, comparing vacuum (εᵣ = 1) vs water (εᵣ ≈ 80), they can see that forces in water are about 80× weaker than in vacuum. The student learns that medium permittivity affects force strength, helping them understand why ionic compounds dissolve in polar solvents.
A user explores potential energy: comparing like charges (U > 0, energy stored) vs opposite charges (U < 0, energy released), they can see how potential energy relates to force. The user learns that potential energy U = kq₁q₂/r represents the work needed to bring charges together, and can see how different charge configurations affect energy. This demonstrates why chemical reactions involving charged species release or absorb energy.
Entering some distances in centimeters and others in meters without converting leads to incorrect forces. The #1 error: entering 5 µC as "5" when the calculator expects coulombs. Since 1 µC = 10⁻⁶ C, you must enter 5×10⁻⁶ C (or 5e-6 in scientific notation). Failing to convert gives results off by a factor of 1,000,000. Always verify units: coulombs (C), microcoulombs (µC = 10⁻⁶ C), nanocoulombs (nC = 10⁻⁹ C), meters (m), centimeters (cm = 0.01 m), millimeters (mm = 0.001 m).
Treating all charges as positive and forgetting to interpret attraction vs repulsion properly. Coulomb's Law formula uses |q₁q₂| (absolute value), which always gives positive force magnitude. But the physical direction—attractive or repulsive—depends on charge signs: opposite signs (+ and − or − and +) attract, same signs (both + or both −) repel. Students often calculate magnitude correctly but misidentify direction, leading to incorrect free-body diagrams or conceptual errors. Always check charge signs to determine force direction.
Assuming force halves when distance doubles (instead of falling to one-quarter). The inverse-square law means F ∝ 1/r², so doubling distance reduces force by 2² = 4×, not 2×. This is a common conceptual error that leads to incorrect predictions about how forces change with distance. Always remember: force decreases with the square of distance, not linearly with distance.
Coulomb's constant k = 8.99×10⁹ N·m²/C² uses meters. Entering distance in centimeters without converting produces force off by 100² = 10,000×. Example: 20 cm must be 0.2 m, not 20. Always convert: 1 cm = 0.01 m, 1 mm = 0.001 m, 1 nm = 10⁻⁹ m. Double-check unit conversions, especially when working with scientific notation.
Entering an εᵣ value but still expecting vacuum-level forces. When charges are in a medium (water, oil, plastic), the medium's molecules partially shield the charges, reducing the force. The effective Coulomb constant becomes k_eff = k/εᵣ. For example, water has εᵣ ≈ 80, so forces in water are ~80× weaker than in vacuum. Forgetting to account for the medium leads to dramatically incorrect force calculations. Always verify that medium settings match your problem conditions.
Applying results directly to extended objects, complex geometries, or real devices where edge effects and materials matter. Coulomb's Law rigorously applies only to point charges—charged objects small enough that their size is negligible compared to separation distance. For extended objects (spheres, rods, planes), you must integrate over the charge distribution using calculus. Using point-charge results for extended objects can lead to significant errors. Always verify that point-charge assumptions are met.
Don't assume this tool is for electrical design or safety compliance—it's for educational purposes only. Real electrical design requires professional analysis, material properties, geometry, safety factors, and regulatory compliance. This tool uses simplified point-charge approximations that ignore these factors. Always consult qualified professionals for electrical design decisions or safety compliance. Understanding limitations helps you use the tool appropriately.
Systematically vary charges and distances to build strong intuition about proportionality and inverse squares. Double charges, halve distances, and observe how force changes. Plot F vs r and F vs 1/r² to verify the inverse-square relationship graphically. This builds deep intuition for why electrostatic forces are strong at short range but negligible at large distances—critical for understanding atomic structure, chemical bonding, and why gravity (not electrostatics) dominates at planetary scales.
Try εᵣ = 1 (vacuum/air) vs higher εᵣ values to see how a dielectric reduces force conceptually. For example, compare forces in vacuum (εᵣ = 1) vs water (εᵣ ≈ 80) vs oil (εᵣ ≈ 2-5). This helps students understand why ionic compounds dissolve in polar solvents—the medium weakens electrostatic attraction between ions, making separation easier. Understanding medium effects helps you understand real-world electrostatic behavior.
Enter off-axis charge positions to practice breaking forces into x and y components. This is essential for understanding how multiple charges create net forces in 2D and 3D space, preparing students for more advanced topics like electric fields and potential. Use coordinate mode to visualize force vectors and understand vector addition in electrostatic contexts.
Electric field E at distance r from charge q is E = kq/r². The force on a test charge q_test is F = q_test × E = q_test × kq/r². This shows Coulomb's Law (F = kq₁q₂/r²) is equivalent to force = charge × field. Use the calculator to find force, then divide by one charge to get the field created by the other—connecting two fundamental electrostatics concepts. Understanding this connection helps you bridge force and field concepts.
Practice finding net forces using the superposition principle. Calculate force from each charge separately, then add them as vectors. This is essential preparation for understanding electric fields from charge distributions and more advanced topics like Gauss's Law. Use superposition mode to analyze complex charge configurations and understand how multiple charges interact.
Create a systematic study plan: Day 1—explore how force changes with charge magnitude. Day 2—explore how force changes with distance. Day 3—explore attractive vs repulsive forces. Day 4—practice with multiple charges. This structured approach builds comprehensive understanding rather than just memorizing formulas. Use the calculator to explore "what-if" scenarios and build deep intuition.
Always remember that this tool is for educational purposes—learning and practice with Coulomb's Law formulas. For engineering applications, consider additional factors like idealized point-charge conditions (no extended objects, complex geometries, or material effects), not an electrical design or safety compliance tool (does not account for real-world factors, safety margins, or regulatory requirements), and real electrical design requires professional analysis. This tool assumes ideal point-charge conditions—simplifications that may not apply to real-world scenarios. For design applications, use professional analysis methods and appropriate safety considerations.
• Point Charge Approximation: Coulomb's Law in its simple form applies to point charges or spherically symmetric charge distributions. Extended charge configurations (line charges, surface charges, complex geometries) require integration techniques not covered by this basic calculator.
• Electrostatic Conditions Only: Calculations assume stationary charges with no time-varying fields. Moving charges produce magnetic fields and electromagnetic interactions described by Maxwell's equations, which are beyond the scope of simple Coulomb analysis.
• Vacuum Permittivity Assumption: The calculator uses free space permittivity (ε₀). Real materials have different dielectric constants that modify electric fields and forces. Conductors, dielectrics, and polarizable media require more sophisticated electrostatic analysis.
• No Quantum Effects: At atomic and subatomic scales, quantum mechanical effects dominate charge interactions. The classical Coulomb model breaks down for electron orbitals, chemical bonding, and nanoscale phenomena where quantum electrodynamics applies.
Important Note: This calculator is strictly for educational and informational purposes only. It demonstrates fundamental electrostatics principles for learning. Real electrostatic design applications—such as ESD protection, capacitor design, or high-voltage engineering—require comprehensive analysis including material properties and safety standards.
The formulas and principles used in this calculator are based on established electrostatics principles from authoritative sources:
Note: This calculator implements point charge approximation formulas for educational purposes. For extended charge distributions, integration methods are required.
Common questions about Coulomb's Law, electrostatic forces, electric charges, and calculations.
Coulomb's Law states that the electrostatic force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them: F = k|q₁q₂|/r². Like charges (both + or both −) repel each other, opposite charges (+ and −) attract. The constant k ≈ 8.99×10⁹ N·m²/C² in vacuum. This means larger charges create stronger forces, and force decreases rapidly as distance increases.
Standard SI units are: Charge in coulombs (C)—note that 1 µC = 10⁻⁶ C, 1 nC = 10⁻⁹ C. Distance in meters (m)—convert cm to m by dividing by 100, mm by dividing by 1000. Force in newtons (N). Coulomb's constant k = 8.99×10⁹ N·m²/C² is calibrated for these units. Mixing units (e.g., entering distance in cm without converting) causes errors by factors of 100² = 10,000×
The inverse-square law (F ∝ 1/r²) arises from geometry: electric field lines spread out uniformly in all directions from a point charge. As you move farther away, the same 'flux' spreads over a sphere of radius r, which has surface area 4πr². The field strength (and hence force) at distance r is diluted by the factor 1/r². Doubling distance spreads the field over 4× the area, reducing force by 4×. This inverse-square behavior is fundamental to both electrostatics and gravity.
Attractive forces pull charges together—this happens when charges have opposite signs (+ and − or − and +). The charges accelerate toward each other. Repulsive forces push charges apart—this happens when charges have the same sign (both + or both −). The charges accelerate away from each other. The formula F = k|q₁q₂|/r² gives magnitude; direction is determined separately by the signs of q₁ and q₂.
Coulomb's Law directly applies to two charges at a time. For systems with more than two charges, use the superposition principle: calculate the force (or field) from each charge separately, then add them as vectors. Many calculators support superposition mode where you input multiple charges and their positions, and the tool computes the net force vector on a test charge by summing individual contributions. This allows analysis of complex charge distributions.
Most Coulomb's Law calculators compute electrostatic force between two charges. However, electric field E and force F are related: E = F/q (field is force per unit charge). To find the electric field created by charge q₁ at a point, use E = kq₁/r². Some calculators have dedicated field mode; otherwise, calculate force on a hypothetical +1 C test charge—the result numerically equals the field magnitude. Fields are measured in N/C or equivalently V/m.
Yes—when charges are immersed in a medium (water, oil, plastic), the medium's molecules partially shield the charges, reducing the force. This is quantified by the relative permittivity εᵣ (dielectric constant). The effective Coulomb constant becomes k_eff = k/εᵣ. For example, water has εᵣ ≈ 80, so forces in water are ~80× weaker than in vacuum. Air has εᵣ ≈ 1.0006 (essentially same as vacuum). This reduction explains why ionic compounds dissolve in polar solvents—the medium weakens electrostatic attraction between ions.
Electric force (F) is the actual force exerted between two specific charges, measured in newtons (N). It depends on both charges: F = kq₁q₂/r². Electric field (E) is the force per unit charge at a point in space, measured in N/C. It's created by one charge and exists even without a second charge present: E = kq/r². The relationship is F = qE, where q is the charge experiencing the field. Fields allow us to analyze forces systematically without tracking every charge pair.
Coulomb's Law rigorously applies only to point charges—charged objects small enough that their size is negligible compared to separation distance. For extended objects (spheres, rods, planes), you must integrate over the charge distribution using calculus. However, for spherically symmetric charge distributions, there's a special case: a uniformly charged sphere acts like a point charge located at its center for external points. This is why planets (spherical masses) obey inverse-square gravity law despite being extended objects.
Compare to familiar forces: (1) Weight of objects—a 1 kg mass weighs ~10 N, so forces of 1-100 N are moderate. (2) Typical magnitudes—everyday static electricity forces are µN to mN (micro- to millinewtons), atomic-scale forces are pN to nN (pico- to nanonewtons). (3) Sign check—opposite charges attract, like charges repel. (4) Distance scaling—halving distance should quadruple force. (5) Unit check—forces >10⁶ N or <10⁻²⁰ N suggest unit conversion errors (forgot µC→C or cm→m).
Electric potential energy U = kq₁q₂/r is the work needed to bring two charges from infinite separation to distance r. It's positive for like charges (repulsive, work required), negative for opposite charges (attractive, energy released). The relationship to force is F = −dU/dr (force is negative gradient of potential energy). As charges move closer together, potential energy of like charges increases (energy stored), opposite charges decreases (energy released). This energy is what's released in chemical reactions involving charged species.
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Select a mode and enter your charge values to calculate electric forces, fields, or potential energy.
F = k · |q₁q₂| / r²
k ≈ 8.99 × 10⁹ N·m²/C² (Coulomb constant)
E = k · q / r²
Field points away from positive, toward negative charges
U = k · q₁q₂ / r
Positive for like charges, negative for opposite charges
F_net = Σ F_i (vector sum)
Net force is the vector sum of individual forces
Coulomb's Law describes the electrostatic force between two charged particles. The force is proportional to the product of the charges and inversely proportional to the square of the distance between them: F = k·|q₁q₂|/r². The force is attractive if charges have opposite signs and repulsive if they have the same sign. The constant k ≈ 8.99×10⁹ N·m²/C² in vacuum.
Electric force (F) is the actual force exerted between two charges, measured in Newtons. Electric field (E) is the force per unit charge at a point in space, measured in N/C or V/m. The field exists even without a test charge present. The relationship is F = qE, where q is the test charge experiencing the field. A positive charge creates a field pointing away from it, while a negative charge creates a field pointing toward it.
The superposition principle states that the net electric force (or field) from multiple charges is the vector sum of the individual forces (or fields) from each charge. Each charge contributes independently, and we add them as vectors. This means we calculate Fₓ and Fᵧ components separately for each charge, then sum them: F_net = F₁ + F₂ + F₃ + ... This principle allows us to analyze complex charge distributions by breaking them into simple two-charge interactions.
Electric potential energy (U = k·q₁q₂/r) is the work needed to bring two charges from infinite separation to distance r. It's positive when like charges (both + or both −) are brought together, meaning work is required to overcome repulsion. It's negative when opposite charges are brought together, meaning the system releases energy as they attract. A system naturally moves toward lower potential energy, so opposite charges spontaneously come together, while like charges naturally separate.
When charges are in a medium other than vacuum (like water or oil), the medium's molecules partially shield the charges, reducing the force. This is quantified by the relative permittivity εᵣ (also called dielectric constant). The effective Coulomb constant becomes k_eff = k/εᵣ. For example, water has εᵣ ≈ 80, so electrostatic forces in water are about 80 times weaker than in vacuum. This is why ionic compounds dissolve in water—the medium significantly reduces the attractive forces between ions.