The Coulomb Potential

Electric potential energy quantifies the work required to assemble a configuration of charges. For two point charges, the energy depends on their magnitudes, signs, and separation. The expression follows directly from Coulomb's law: $U=\frac{\mathrm{k\; q\_1\; q\_2}}{r}$, where $k$ is Coulomb's constant, approximately 8.9875 × 10⁹ N·m²/C². The sign of the energy reflects whether work must be done against electric repulsion or is released during attraction. Positive energy indicates repulsive configurations, while negative energy corresponds to bound states such as electron–proton interactions in the hydrogen atom.

Derivation from Coulomb's Law

Coulomb's law gives the force between two point charges, . Potential energy is the integral of force with respect to separation: $U=-\backslash int\_\{\backslash infty\}^\{r\}\; F\backslash ,dr$. Evaluating the integral yields the familiar inverse relationship with distance. This formulation assumes the energy reference point at infinite separation, so assembling the charges from infinity to distance r either requires or releases energy depending on their signs. The inverse dependence on r highlights how bringing charges closer dramatically increases interaction strength.

Using the Calculator

Enter the charges in coulombs. To compute potential energy, supply their separation distance and choose the appropriate mode. The script evaluates $U=\frac{\mathrm{k\; q\_1\; q\_2}}{r}$. To solve for distance given energy, provide U instead; rearranging yields $r=\frac{\mathrm{k\; q\_1\; q\_2}}{U}$. Negative energy values correspond to attractive configurations; the calculator handles signs appropriately.

Visualization and Interpretation

Plotting potential energy as a function of separation produces a hyperbola with an asymptote at zero energy as r approaches infinity. For unlike charges, the curve dips below the axis, reflecting the energy released when charges attract. For like charges, the curve lies above the axis, signifying the work needed to compress the charges together. Such plots help students see how small changes in distance dramatically alter energy, a concept important in chemistry, atomic physics, and electrical engineering.

Worked Example

Consider bringing a +2 µC charge within 0.03 m of a −3 µC charge. Substituting into the formula gives $U=\frac{\mathrm{8.9875\backslash times10^\{9\}\; \backslash times\; 2\backslash times10^\{-6\}\; \backslash times\; (-3\backslash times10^\{-6\})}}{0.03}$ ≈ −1.8 J. The negative sign indicates energy is released as the opposite charges come together. If the charges were both positive, the result would be +1.8 J, meaning external work is required to maintain the separation.

Sample Values

q₁ (µC)	q₂ (µC)	r (m)	U (J)
1	1	0.10	0.09
2	−3	0.05	−2.16
5	4	0.20	0.90
−7	3	0.15	−1.26

Physical Significance

Electric potential energy plays a key role in electrostatics, chemistry, and particle physics. The energy stored in capacitor plates, for example, arises from the potential energy of separated charges. In molecules, potential energy between electrons and nuclei determines bond strength and reaction dynamics. Even macroscopic phenomena like lightning involve vast amounts of electrostatic potential energy converted to kinetic and thermal energy during discharge.

Historical Perspective

The concept of potential energy emerged in the 19th century as physicists like Joule and Helmholtz recognized energy conservation across mechanical and electromagnetic systems. Coulomb’s meticulous torsion balance experiments provided quantitative measurements of the force between charges, laying the groundwork for the energy formulation used today. Later, the development of field theory and Maxwell’s equations broadened the understanding of how energy is stored and transmitted in electromagnetic fields.

Limitations and Extensions

The point-charge model fails when charges occupy finite volumes or when quantum effects become significant. At extremely small separations, quantum electrodynamics replaces classical Coulomb potential with more nuanced interactions. In media with dielectric constants different from vacuum, the effective constant becomes $\mathrm{k/\backslash varepsilon\_r}$, reducing energy by a factor equal to the material's relative permittivity. These refinements extend the basic concept to real-world materials and high-energy physics.

Relation to Other Laws

Electric potential energy links closely with Gauss's law and electric field concepts. The gradient of potential energy with respect to position yields the force, connecting energy and dynamics. In circuits, potential energy differences correspond to voltage. The notion also parallels gravitational potential energy, highlighting the unifying theme of inverse-square forces across physics.

Conclusion

By quantifying the energy between charges, this calculator helps students and practitioners analyze electrostatic interactions quickly. The extensive discussion above situates the simple formula within a broader theoretical and practical framework, encouraging deeper exploration of electrostatics.