Dipole–Dipole Interaction Calculator

Dipole–dipole interaction energy for separation vector r⃗ is:

• k = 1/(4π ε0 εr) for electric dipoles.
• k = μ0 μr/(4π) for magnetic dipoles.
• Field from dipole 1 at dipole 2 uses F⃗ = k(3(p⃗·r̂)r̂ − p⃗)/r^3.
• Torque on dipole 2: τ⃗ = p⃗2 × F⃗ (electric uses E, magnetic uses B).
• Force is computed numerically as F⃗ = −∇U via central differences.

1. Select electric or magnetic interaction type.
2. Choose dipole units and separation units.
3. Pick an input mode for each vector.
4. Enter dipole 1, dipole 2, and separation values.
5. Set εr or μr to match your medium.
6. Press Compute to see results above the form.
7. Use CSV or PDF buttons to export the run.

1) What this calculator models

This tool evaluates the classic point-dipole interaction between two dipole moments separated by a vector r⃗. It reports interaction energy, the field from dipole 1 at dipole 2, the torque on dipole 2, and the force obtained from the energy gradient. Results are computed in SI units with clear unit conversions.

2) Why orientation matters

Dipole coupling is strongly anisotropic. For electric dipoles, head-to-tail alignment along r⃗ is typically attractive, while side-by-side parallel alignment is typically repulsive. Rotating either dipole changes the dot products p⃗1·p⃗2 and p⃗i·r⃗, which can flip the sign of U and create substantial torque even when the separation distance is fixed.

3) Distance scaling you can sanity-check

The leading interaction energy scales as 1/r³. Doubling the separation reduces |U| by about eight times, and the field magnitude also scales as 1/r³ for a fixed dipole. This steep scaling is why dipole effects dominate at nanometer to micrometer distances, yet can become negligible at macroscopic separations in many laboratory setups.

4) Medium parameters and coupling constant k

The calculator uses k = 1/(4π ε0 εr) for electric dipoles and k = μ0 μr/(4π) for magnetic dipoles. Increasing εr screens electric interactions: for example, moving from vacuum (εr≈1) to water (εr≈78 at room temperature) reduces energy and field by roughly 78× for the same geometry.

5) Electric dipole magnitudes with example numbers

Molecular electric dipoles often range from 0.1 to 10 Debye. A useful benchmark is two 10 D dipoles separated by 2 nm in a medium with εr=2, aligned head-to-tail along r⃗. The point-dipole model gives U ≈ −1.25×10⁻²¹ J, which is about −7.8 meV or −0.75 kJ/mol, indicating weak but measurable coupling.

6) Magnetic dipoles and practical scales

For atomic-scale moments, the Bohr magneton μB is a natural unit. A single μB at ångström separations produces very small energies, but collections of moments in nanoparticles can yield much larger effective dipoles. When μr≈1, magnetic dipole interactions are often weaker than comparable electric cases unless the magnetic moment is large or separations are extremely small.

7) Reading field, torque, and force outputs

The field vector shown is produced by dipole 1 at the position of dipole 2. The torque τ⃗ = p⃗2×E⃗ (or p⃗2×B⃗) tends to rotate dipole 2 toward alignment with the local field. The force output comes from a central-difference approximation to −∇U, so it is most reliable when r⃗ is not extremely small.

8) Modeling tips and limitations

Point-dipole formulas assume the dipole size is much smaller than r. If your dipoles are extended objects, near-field corrections may be needed. Use consistent coordinate conventions for θ and φ, and compare trends rather than single numbers when exploring parameter sweeps. For very small r, the 1/r³ behavior can overestimate coupling in real materials.

1) What does a negative interaction energy mean?

Negative U indicates an energetically favorable configuration relative to infinite separation. For many geometries, head-to-tail alignment produces negative U, while side-by-side parallel alignment produces positive U, but orientation and medium parameters can change this.

2) Why is the force computed numerically?

The force is F⃗ = −∇U with respect to the separation vector. A robust central-difference gradient works for any orientation without lengthy symbolic expressions, and it stays consistent with the energy formula used in the calculator.

3) Which angle conventions are used for θ and φ?

θ is the polar angle measured from the +z axis. φ is the azimuth measured from the +x axis in the x–y plane. This is the standard physics spherical-coordinate convention used to convert magnitude and angles into vector components.

4) How do εr and μr affect the results?

Electric results scale inversely with εr, so larger εr reduces energy and field. Magnetic results scale proportionally with μr, so larger μr increases coupling. Many common non-magnetic media have μr very close to 1.

5) When does the point-dipole approximation fail?

It can fail when the physical dipole size is not much smaller than the separation, or when charge or magnetization distributions overlap. In such cases, finite-size or multipole models may be needed for accurate near-field predictions.

6) Why do I see large values at very small separations?

Because the formulas scale as 1/r³, shrinking r rapidly increases magnitude. Real systems often saturate or deviate due to finite size, quantum effects, or material response, so treat extremely small-r outputs as idealized.

7) Can I use this for quick parameter sweeps?

Yes. Keep one dipole fixed, vary r or orientation angles, and export each run via CSV or PDF. For systematic sweeps, record consistent units and compare the predicted 1/r³ scaling as a sanity check.