The Curie's law calculator above applies the relation χ = C/T, and its Curie–Weiss extension χ = C/(T − θ), to a paramagnetic material. Give it a Curie constant, a temperature and an applied field, and it returns the volume susceptibility, the magnetisation that field induces, the flux density that magnetisation contributes and the relative permeability the material presents. It also runs the other way, turning a measured susceptibility into the temperature it implies.
Arb Digital keeps one job per page. This tool consumes a Curie constant. It does not derive one. If you need to work out C in the first place from an ion's effective magnetic moment and the number of moment-carrying ions per unit volume, that is the job of the Curie constant calculator, and the two pages are designed to be used in sequence rather than in competition.
What This Curie's Law Calculator Does
Curie's law states that the magnetic susceptibility of a paramagnet is inversely proportional to absolute temperature. The physical picture is a competition. An applied field lowers the energy of moments pointing along it, giving them a statistical preference for alignment. Thermal agitation constantly knocks them out of alignment again. Raise the temperature and thermal energy wins by a wider margin, so the same field produces less net alignment and the susceptibility falls.
Susceptibility is a ratio, not a magnetisation. It tells you how much magnetisation you get per unit of applied field. The tool therefore reports the susceptibility as the headline figure and the magnetisation separately, because the two answer different questions and mixing them up is one of the more persistent confusions in the subject.
The grid shows the magnetisation in amperes per metre, the flux density that magnetisation contributes in tesla, the relative permeability — which for a paramagnet is 1 plus a very small number — and the applied field converted to H so you can see what the tool actually used.
How to Use It
- Get a Curie constant first. Either from a measurement, from a published table on the matching basis, or from the companion calculator that derives it from the ion's moment and density.
- Work in kelvin. The law is about absolute temperature. Entering 20 for room temperature because you were thinking in Celsius gives an answer fifteen times too large.
- Set the Weiss offset deliberately. Leave it at zero unless you have a reason. A non-zero θ is a statement that the moments interact, and it should come from the intercept of a measured 1/χ plot.
- Choose the field unit that matches your instrument. Magnet specifications are usually in tesla or gauss; coil calculations produce amperes per metre. The tool converts either way.
- Sanity-check the magnetisation. If it approaches the saturation magnetisation of the material, the linear law has stopped applying and the Brillouin function is required instead.
The Formula: How Curie's Law Is Calculated
The basic form is χ = C ÷ T, with T in kelvin and χ dimensionless on the volume basis. The Curie–Weiss generalisation is χ = C ÷ (T − θ), where θ is the Weiss temperature that accounts for interactions between moments. Setting θ to zero recovers the original law exactly.
Magnetisation follows as M = χH, and the flux density the material itself contributes is μ0M. Relative permeability is μr = 1 + χ. When a flux density is supplied instead of a field strength, the tool converts with H = B ÷ μ0, which is accurate for a weak paramagnet because the sample's own contribution to B is negligible next to the applied field.
The temperature dependence comes from the Boltzmann factor in the underlying statistics, weighing magnetic energy against kBT. That is where the exact NIST CODATA value for the Boltzmann constant of 1.380649 × 10−23 J/K enters, although in the finished law it has already been absorbed into C.
Work the defaults through by hand. With C = 4.93 K, θ = 0 and T = 300 K, the susceptibility is 4.93 ÷ 300 = 0.016433. A 1 T flux density corresponds to H = 1 ÷ 1.256637 × 10−6 = 795,775 A/m, so the magnetisation is 0.016433 × 795,775 = 13,077 A/m and the material's own flux contribution is 1.256637 × 10−6 × 13,077 = 0.01643 T, or 16.4 mT. The relative permeability is 1.016433.
Why the Weiss Offset Matters More Than It Looks
A Weiss temperature is not a fudge factor. It is a measurement of how strongly the moments feel each other. A positive θ means exchange favours parallel alignment, so the material helps the applied field and susceptibility is larger than pure Curie behaviour would give at the same temperature. A negative θ means the coupling favours antiparallel alignment and susceptibility is suppressed.
The consequence is dramatic close to a positive θ. Ten kelvin above a Weiss temperature of 290 K, the denominator is 10 rather than 300, so the susceptibility is thirty times larger than the naive law predicts. This is the divergence that signals an approaching ferromagnetic transition, and it is why a material can look almost inert at room temperature and then behave dramatically a few degrees lower.
Below θ the formula returns a negative number, which is not a physical susceptibility. It is the model telling you it has left its domain. Below the ordering temperature the material has spontaneous magnetisation, the response is hysteretic, and a single susceptibility no longer describes it. The tool says so rather than printing the negative value as if it meant something.
Susceptibility Conventions That Trip People Up
Three susceptibilities circulate under the same symbol. Volume susceptibility is dimensionless in SI and is what this page uses. Mass susceptibility divides that by density and carries cubic metres per kilogram. Molar susceptibility multiplies the mass form by molar mass. They differ by many orders of magnitude, and a Curie constant is only meaningful alongside the susceptibility basis it was derived on.
The SI-versus-CGS split is worse, because in CGS the volume susceptibility is also called dimensionless yet differs from the SI value by a factor of 4π. Older tables in emu are extremely common in chemistry and geology. Before trusting any published C, check which system and which basis it belongs to. HyperPhysics on the magnetic properties of solids sets out how diamagnetic, paramagnetic and ferromagnetic responses are classified, and OpenStax University Physics Volume 2, section 12.7 on magnetism in matter, gives the SI treatment.
Where Curie's Law Stops Being Valid
The law is a linear approximation valid when the magnetic energy of a moment in the field is much smaller than its thermal energy. That condition is easy to satisfy at room temperature and easy to break at low temperature. In a superconducting magnet at a few kelvin, a paramagnetic salt can be driven well towards saturation, at which point the magnetisation stops rising linearly with field and the susceptibility becomes field dependent. The correct description there is the Brillouin function, of which Curie's law is the small-argument limit.
Two other contributions sit alongside the paramagnetic term in any real measurement. Every material has a small negative diamagnetic susceptibility from the orbital response of paired electrons, which is temperature independent and can dominate in a weakly paramagnetic sample. Metals add Pauli paramagnetism from conduction electrons, which is also nearly temperature independent. A measured susceptibility is the sum, so fitting C/T to raw data without subtracting a constant background gives a Curie constant that is systematically wrong.
Demagnetising fields are the practical trap. The field inside a sample is not the field you applied unless the sample is a long thin rod aligned with the field. For a sphere or a disc the shape correction matters once the susceptibility stops being tiny, and it is geometry, not physics, that decides the size of the correction.
How This Sits Next to the Other Magnetism Tools
Upstream of this page is the Curie constant calculator, which derives C from the effective moment and carrier density. Alongside it, the magnetic field converter moves values between tesla, gauss and amperes per metre, and the temperature converter gets you into kelvin before the arithmetic starts.
To know what field your apparatus can apply in the first place, the solenoid magnetic field calculator covers a coil and the magnetic field of a wire calculator covers a straight conductor, while the solenoid inductance calculator handles the coil's own inductance. For the thermal statistics that sit underneath the law, the Boltzmann factor calculator shows how strongly a temperature suppresses an energy level.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using Celsius instead of kelvin — the law is about absolute temperature, and the error near room temperature is a factor of fifteen.
- Pairing a molar Curie constant with a volume susceptibility — the two bases differ by the molar volume and cannot be mixed.
- Fitting raw susceptibility data without a background — diamagnetic and Pauli contributions are temperature independent and must be subtracted before fitting.
- Trusting the result below the Weiss temperature — the formula returns a negative number there, which means the model has left its domain rather than that susceptibility is negative.
- Ignoring demagnetising fields — only a long rod aligned with the field sees the applied field internally; other shapes need a geometry correction.
Related Free Tools From Arb Digital
Pair this with the Curie constant calculator, which supplies the C this page needs. The magnetic field converter and temperature converter handle units, the solenoid magnetic field calculator and magnetic field of a wire calculator tell you what field you can apply, and the Boltzmann factor calculator covers the underlying thermal statistics. The solenoid inductance calculator completes the coil side. Everything is on the free online tools hub.
Frequently Asked Questions
This page uses a Curie constant you already have and returns a susceptibility or magnetisation at a temperature and field. The Curie constant calculator works in the other direction, deriving that constant from the effective magnetic moment of an ion and the number of such ions per unit volume. One consumes the constant, the other produces it.
It is the offset in the Curie-Weiss form, chi equals C over T minus theta, and it measures how strongly moments interact. Positive values indicate exchange favouring parallel alignment, negative values antiparallel coupling. Take it from the intercept of a measured plot of reciprocal susceptibility against temperature rather than guessing it.
Because the denominator T minus theta changes sign once the temperature drops below the Weiss temperature. That is the model signalling that it has left its range of validity, not a real negative susceptibility. Below the ordering temperature a paramagnetic description no longer applies and the response becomes hysteretic.
No. Susceptibility is the ratio of magnetisation to applied field and is dimensionless on the volume basis. Magnetisation is a magnetic moment per unit volume, measured in amperes per metre, and depends on the field you actually applied. The tool reports both because they answer different questions.
Yes. A value in tesla or gauss is a flux density, and the tool divides it by the vacuum permeability to get the field strength H that the relation needs. That conversion is accurate for a weak paramagnet, where the material's own contribution to the flux density is negligible compared with the applied field.
No. Diamagnetism comes from the induced orbital response of paired electrons, is negative and is almost independent of temperature. Curie's law describes permanent moments being partly aligned against thermal agitation. A real measurement contains both, so a diamagnetic background usually has to be subtracted before fitting.
When the magnetic energy of a moment in the field stops being small compared with its thermal energy, which happens at high field and low temperature. The magnetisation then approaches saturation and stops rising in proportion to the field. The Brillouin function describes the whole range, and Curie's law is its small-argument limit.
This tool is provided for educational use. It applies the ideal paramagnetic result, ignores diamagnetic and Pauli backgrounds, demagnetising-field corrections and saturation, and is not valid at or below an ordering temperature.
