Calculator
Formula used
For a magnetic dipole moment m in a uniform magnetic field B, the potential energy is:
U = − m B cos(θ)
In full vector form, the same relationship is:
U = − (m · B)
- U is in joules (J).
- m is in A·m² (equivalently J/T).
- B is in tesla (T).
- θ is the angle between m and B.
How to use this calculator
- Select Magnitude & Angle or Vector Components.
- Choose units for dipole moment and magnetic field.
- Enter values in the visible input boxes.
- Press Calculate to show results above the form.
- Use Download CSV or Download PDF after computing.
Example data table
| m (A·m²) | B (T) | θ (deg) | U (J) | Meaning |
|---|---|---|---|---|
| 0.25 | 0.80 | 0 | −0.200000 | Aligned, lowest energy. |
| 0.25 | 0.80 | 90 | 0.000000 | Perpendicular, zero interaction energy. |
| 0.25 | 0.80 | 180 | 0.200000 | Opposed, highest energy. |
| 1.00 | 0.05 | 30 | −0.043301 | Partially aligned, moderate negative energy. |
Tip: Negative values indicate stable alignment with the field direction.
Magnetic dipole potential energy in practice
1) What this calculator quantifies
The tool evaluates magnetic potential energy, U, for a dipole moment interacting with a uniform field. It reports sign and magnitude in joules, plus convenient mJ and µJ conversions. This helps compare alignment stability across different dipole strengths and field levels.
2) Why the sign of U matters
When U is negative, the configuration is energetically favorable and the dipole tends to align with the field direction. Positive values indicate opposition to the field, requiring work to maintain orientation. At θ = 90°, the cosine term is zero and the interaction energy vanishes.
3) Typical magnetic field levels
Many lab scenarios span wide field ranges. Earth’s ambient field is on the order of tens of microtesla, while small coils can reach millitesla levels with modest currents. Strong permanent magnets near their surface can approach fractions of a tesla, making dipole alignment effects clearly measurable.
4) Typical dipole moment scales
Dipole moments vary by system: tiny sensing elements can be in the milliampere–square‑meter range, while macroscopic magnet assemblies can be orders of magnitude larger. Because U scales linearly with both m and B, doubling either input doubles the predicted energy change.
5) Energy differences you can compare
A useful metric is the change between aligned and anti‑aligned states: ΔU = U(180°) − U(0°) = 2mB. This directly represents the energy barrier to flip a dipole in a fixed field. Larger mB products imply stronger restoring torque and greater stability.
6) Component mode for real geometry
If you know orientation through coordinates, the component mode uses U = −(m · B). This is ideal for simulations, sensor calibration, and experiments where the field has known components (for example, a Helmholtz coil pair). The calculator also estimates the angle between vectors when magnitudes are nonzero.
7) Units and reporting consistency
The calculator converts common entries to SI internally. Dipole moment uses A·m² (equivalently J/T), and field uses tesla. Gauss is supported for field conversion. Keeping results in joules allows direct comparison with mechanical work and thermal energy scales in related analyses.
8) Practical interpretation and limits
This model assumes a uniform field and a rigid dipole moment. In strongly nonuniform fields, spatial gradients can create net forces in addition to torques, and effective moments may vary with material properties. Use the output as a clean baseline for energy and alignment studies.
FAQs
1) What is the magnetic potential energy of a dipole?
It is the interaction energy between a magnetic dipole moment and an external magnetic field, commonly computed as U = −mBcos(θ) or U = −(m · B).
2) Why does the formula have a negative sign?
The negative sign indicates that energy decreases when the dipole aligns with the field. Alignment is energetically favorable, while anti‑alignment increases potential energy and is less stable.
3) What does U = 0 mean in this context?
Zero interaction energy typically occurs when the dipole is perpendicular to the field (θ = 90°) or when the external field magnitude is zero. In these cases, the dot product is zero.
4) Can I use negative field or negative dipole inputs?
Use magnitudes as positive values in magnitude‑angle mode. If direction matters, use component mode, where signs in vector components naturally represent direction and determine the dot product.
5) What units should I use for dipole moment?
The SI unit is A·m², which is equivalent to J/T. This calculator accepts A·m², J/T, mA·m², and emu (erg/G), converting internally to A·m² for computation.
6) How do I interpret a positive result?
A positive U means the dipole is oriented against the field direction, which is energetically unfavorable. The system tends to rotate toward lower energy, producing a restoring torque.
7) Does this calculator account for field gradients and forces?
No. It models energy in a uniform field and focuses on torque‑related alignment. In nonuniform fields, additional force terms can appear due to spatial gradients, requiring a more detailed magnetic model.