Calculate electric forces, fields, and potential energy between charged particles with vector analysis and superposition.
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
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.
Consider Emma, a physics student tackling her electrostatics homework: "Two charges of +3 µC and -5 µC are separated by 0.2 meters. What is the force between them?" She knows the formula involves charges and distance, but which formula? How do the signs affect the result? And critically, how do you convert microcoulombs (µC) to coulombs (C) without losing track of powers of ten? Using a Coulomb's Law calculator, she enters q₁ = 3×10⁻⁶ C, q₂ = -5×10⁻⁶ C, and r = 0.2 m. The calculator instantly shows: Force = 3.37 N (attractive). More importantly, it displays the step-by-step calculation and explains that opposite charges attract, helping Emma verify her manual work and build intuition for how charge magnitude, distance, and signs interact.
This calculator helps you solve core electrostatics problems: (1) Coulomb's Law—calculate the electrostatic force between two point charges given their magnitudes, signs, and separation distance. (2) Electric field—determine the field created by a charge at a specific point in space. (3) Potential energy—find the energy stored in a system of charges. (4) Superposition—analyze net forces when multiple charges interact simultaneously. Whether you're checking homework, preparing for exams, exploring inverse-square relationships, or teaching electrostatics concepts, this tool transforms abstract formulas into concrete, verifiable results.
Coulomb's Law is the basic law of electrostatics linking electric charge to force, similar to how Newton's law of gravitation links mass to gravitational force. It quantifies how the force between two charges depends on their magnitudes, their signs (positive or negative), and the distance separating them. The tool implements the standard point-charge Coulomb's Law, making it an ideal teaching and learning companion for introductory physics and engineering courses.
Common questions users have include: "How strong is the electric force between two charges?" "How does the force change if I double the distance or double the charge?" "How does the medium (air vs vacuum) affect the force?" This calculator helps you quickly plug in values from physics problems or lab setups, see how the force scales as charges or distance change, and build intuition about attractive vs repulsive interactions. At this level, it is a teaching and learning companion, not a full electrostatic design simulator.
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. The tool lets you enter charges with sign and automatically converts/scales to the appropriate units for force calculations.
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. Typically meters in SI; the calculator may accept other units and convert internally.
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, as exposed in the UI. 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.
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.
In scalar mode, the calculator shows only the magnitude of the force (N). In vector mode, it displays the direction along the line joining charges and may include components if coordinates are entered. The calculator may allow coordinate-based inputs and show vector components conceptually, helping you practice 2D vector decomposition of Coulomb forces for physics and engineering classes.
Important reminders: Inputs should be consistent in units (for example, all distances in meters). The calculator is built for point charges and conceptual learning; real objects have size, shape, and material effects that are not modeled here.
Or, using permittivity: F = (1 / (4π ε)) · |q₁ q₂| / r², where ε = εᵣ × ε₀. F is force magnitude in newtons, q₁ and q₂ are charges in coulombs, r is distance in meters. The absolute value ensures force magnitude is always positive; direction (attractive vs repulsive) is determined separately by the signs of q₁ and q₂.
Attractive vs repulsive: Multiply q₁ and q₂; the sign tells you whether the force tends to pull charges together or push them apart. The force points along the line connecting charges, with direction depending on charge signs. Opposite charges (+ and −) attract, pulling them together. Like charges (both + or both −) repel, pushing them apart.
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.
Example 1 — Simple Two-Charge Problem:
Calculation:
F = (8.99×10⁹ N·m²/C²) × |(2×10⁻⁶ C)(−3×10⁻⁶ C)| / (0.5 m)²
F = (8.99×10⁹) × |−6×10⁻¹² C²| / 0.25 m²
F = (8.99×10⁹) × (6×10⁻¹² / 0.25)
F = (8.99×10⁹) × (2.4×10⁻¹¹)
F = 0.216 N (attractive)
Direction: Attractive because charges have opposite signs (+ and −).
Example 2 — Distance Effect:
Same charges as above (q₁ = +2.0 µC, q₂ = -3.0 µC), but r doubled to 1.0 m.
F = (8.99×10⁹) × |(2×10⁻⁶)(−3×10⁻⁶)| / (1.0 m)²
F = (8.99×10⁹) × (6×10⁻¹²) / 1.0
F = (8.99×10⁹) × (6×10⁻¹²)
F = 0.054 N (attractive)
The force drops by a factor of 4 (0.216 / 0.054 = 4), demonstrating the inverse-square relationship: doubling distance reduces force by 2² = 4×.
A student enters values from a textbook question into the calculator to check their manual Coulomb's Law calculation and better understand scaling with distance. For example, a homework problem asks: "Find the force between charges of +4 µC and +6 µC separated by 0.3 m." The student calculates manually: F = 8.99×10⁹ × (4×10⁻⁶)(6×10⁻⁶) / (0.3)² = 2.40 N. Using the calculator to verify: enter q₁ = 4e-6 C, q₂ = 6e-6 C, r = 0.3 m → Result: 2.40 N (repulsive) ✓. The calculator confirms the answer and reminds that like charges repel, building confidence and catching unit conversion errors before submitting homework.
In an electrostatics lab with charged pith balls or small spheres, a student uses the calculator to estimate expected force for given charges and separations. Before the lab, they calculate theoretical forces to compare with measured values, helping identify experimental errors or equipment issues. The calculator provides quick "what-if" scenarios: "What if the charges are 10× larger?" or "What if the distance is halved?" This preparation builds intuition and helps students understand whether their experimental results are reasonable.
A physics teacher uses the calculator to quickly generate several scenarios for classroom examples, showing how changing q or r affects F. For instance, they demonstrate the inverse-square law by fixing charges and systematically varying distance: r = 0.1 m → F = 899 N, r = 0.2 m → F = 225 N, r = 0.4 m → F = 56 N. Students see that doubling distance reduces force by 4×, making the abstract relationship concrete and memorable. The calculator helps teachers create engaging, interactive lessons that build deep understanding.
A curious learner compares the electric force between two small charges with the gravitational force between the same masses (using Force / Work / Power or other tools for the gravity part), to see how much stronger electrostatic interactions can be. For example, comparing the electric force between two electrons with their gravitational attraction reveals that electric forces are about 10⁴² times stronger at atomic scales. This comparison helps students understand why electrostatic forces dominate at atomic and molecular scales, while gravity dominates at planetary scales.
An engineering student revising for exams uses the tool to practice multiple-charge and vector problems as a stepping stone to more complex field and potential calculations. They work through problems involving three or more charges, using superposition to find net forces, then connect these results to electric field concepts. The calculator helps bridge the gap between basic Coulomb's Law and advanced topics like Gauss's Law, capacitance, and circuit analysis.
In NaCl (table salt), a Na⁺ ion (+1e = +1.6×10⁻¹⁹ C) and a Cl⁻ ion (−1e = −1.6×10⁻¹⁹ C) are separated by about 0.28 nm = 2.8×10⁻¹⁰ m. Calculate the attractive force: F = 8.99×10⁹ × (1.6×10⁻¹⁹)² / (2.8×10⁻¹⁰)² ≈ 2.9×10⁻⁹ N. While tiny by everyday standards, this force is enormous at the atomic scale, holding the ionic crystal together. The calculator helps quantify why ionic compounds have high melting points and are stable solids at room temperature.
To understand how distance affects force, test systematically: Fix charges at q₁ = q₂ = 1 µC, vary distance: r = 0.1 m → F = 899 N, r = 0.2 m → F = 225 N, r = 0.4 m → F = 56 N. Doubling distance (0.1→0.2) reduces force by 4× (899/225 ≈ 4), doubling again (0.2→0.4) reduces by 4× (225/56 ≈ 4). This demonstrates the inverse-square law viscerally: force decreases rapidly with distance, which explains why electrostatic forces dominate at atomic scales but are negligible at macroscopic distances.
A student keeps a log of different Coulomb's Law problems solved using the calculator, noting how force changes with different charge combinations and distances. They create a personal reference guide showing patterns: "When I double both charges, force quadruples. When I double distance, force quarters." This systematic exploration builds strong intuition and helps with exam preparation, as students can quickly estimate answers before doing detailed calculations.
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).
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.
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.
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.
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.
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.
Using point-charge results as if they were safe design thresholds for equipment or insulation. This calculator is built for educational purposes and conceptual learning. Real-world electrical design requires professional engineering analysis, accounting for material properties, geometry, safety factors, and regulatory standards. Never use calculator results for actual equipment design or safety-critical applications.
Failing to verify that calculated forces are physically reasonable. Compare to familiar forces: a 1 kg mass weighs ~10 N, so forces of 1-100 N are moderate. Everyday static electricity forces are µN to mN (micro- to millinewtons), atomic-scale forces are pN to nN (pico- to nanonewtons). Forces >10⁶ N or <10⁻²⁰ N suggest unit conversion errors (forgot µC→C or cm→m). Always do a sanity check before accepting results.
Encourage users to double charges, halve distances, and observe how the force changes to build strong intuition about proportionality and inverse squares. Systematically vary distance while keeping charges constant to see 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.
Suggest trying 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.
Recommend entering 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.
Suggest using Kinetic & Potential Energy, Force / Work / Power, or Electric fields concepts (if available) to connect Coulomb's Law to energy and motion problems conceptually. For example, calculate the force between charges, then use work-energy principles to find how much energy is required to move charges apart. This connects force concepts to energy concepts, building a more complete understanding of electrostatics.
Encourage creating your own practice questions, then using the calculator to check solutions. For example: "What if I triple both charges and halve the distance? How does force change?" Answer: Force increases by 3² × 2² = 9 × 4 = 36×. This active problem-creation helps students internalize the relationships and prepares them for exams.
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.
If the calculator supports multiple charges, 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.
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.
The calculator computes the electrostatic force between two point charges using Coulomb's Law: F = k|q₁q₂|/r², where k ≈ 8.99×10⁹ N·m²/C². It can also calculate electric fields, potential energy, and net forces from multiple charges using the superposition principle. The tool handles unit conversions, determines whether forces are attractive or repulsive based on charge signs, and provides step-by-step calculations for educational purposes.
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 calculator may accept various units and convert internally, but always verify your inputs match the expected format.
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. The calculator automatically labels the interaction as attractive or repulsive based on the charge signs you enter.
No, this calculator uses the point-charge approximation, which assumes charges are small compared to their 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. Real materials have complex geometries, edge effects, and material properties that are not modeled here. This tool is designed for educational and conceptual learning, not detailed device design.
Coulomb's constant k = 8.99×10⁹ N·m²/C² is related to the permittivity of free space ε₀ by k = 1/(4πε₀). The formula F = k|q₁q₂|/r² is equivalent to F = (1/(4πε))|q₁q₂|/r², where ε = εᵣ × ε₀. Using k is convenient for vacuum/air calculations, while using ε allows easy incorporation of different media through the relative permittivity εᵣ. Both approaches give the same result; it's a matter of mathematical convenience and the context of the problem.
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 and is essential for understanding electric fields from multiple sources.
The calculator provides accurate results for point charges and spherically symmetric charge distributions (when viewed from outside). For extended objects with complex geometries, the results are approximate and should be used for conceptual understanding only. Real-world accuracy requires professional engineering analysis accounting for material properties, geometry, edge effects, and safety factors. This tool is designed for educational purposes, homework checking, and building intuition—not for actual equipment design or safety-critical applications.
Yes! This calculator is designed specifically for educational use, including homework checking, exam preparation, and conceptual learning. It helps you verify manual calculations, understand how force scales with charge and distance, and build intuition about electrostatic interactions. However, always show your work and understand the underlying physics—don't just copy calculator results. Use the tool to check your answers and learn from mistakes, not as a substitute for understanding the concepts.
Mixing units causes dramatic errors. For example, entering distance in centimeters when the calculator expects meters produces force off by 100² = 10,000×. If you enter 20 cm as "20" instead of "0.2" meters, your force will be 10,000 times too large. Always convert to consistent SI units before entering values: 1 cm = 0.01 m, 1 mm = 0.001 m, 1 µC = 10⁻⁶ C, 1 nC = 10⁻⁹ C. Double-check your unit conversions, especially when working with scientific notation.
Yes! 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. Electric potential energy U = kq₁q₂/r is the work needed to bring charges from infinite separation to distance r. The calculator can compute these related quantities, helping you connect force, field, and energy concepts in electrostatics.
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, and understanding it is crucial for physics and engineering.
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, making separation easier.
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