The field energy density calculator above answers a question that is easy to state and surprisingly hard to find a clean answer to: how much energy is actually sitting in a field? Not in the capacitor, not in the coil, but in the space itself. Both an electric field and a magnetic field carry energy in every cubic metre they occupy, and this page reports both densities, their sum, and the total energy across a volume you nominate.
Arb Digital builds free physics calculators that each own one job. The capacitor energy calculator gives the total joules in a component from its capacitance and voltage; this page works one level down, at the field itself, so it applies equally to a capacitor gap, a solenoid bore, a transmission line, or a patch of empty space with a radio wave passing through.
What This Field Energy Density Calculator Does
Energy density is energy per unit volume, measured in joules per cubic metre. For an electric field the density is one half the permittivity times the field squared. For a magnetic field it is the flux density squared divided by twice the permeability. Where both fields are present, the two densities simply add — there is no cross term, which is a genuinely useful fact and not an approximation.
The tool computes both terms, sums them, and multiplies by the volume you give to produce a total in joules. It also reports what fraction of the stored energy is magnetic, because in almost every practical situation that fraction is either very close to zero or very close to one, and knowing which tells you immediately what kind of system you are dealing with.
The relative permittivity and permeability inputs let you move out of vacuum. Filling a capacitor gap with a dielectric of relative permittivity ten multiplies the electric energy density by ten at the same field strength. Filling a magnetic circuit with iron does something similar but far less linearly, because iron's permeability is a strong function of the field itself and collapses toward one at saturation.
How to Use It
- Enter the field strengths you have. Leave either one at zero if your situation is purely electric or purely magnetic. The tool handles both cases without complaint.
- Set the relative permittivity or permeability if you are not in vacuum or air. Air is close enough to one for both that the difference rarely matters.
- Give the volume the fields occupy. This is where care is needed: the formulas assume the field is uniform across the volume, which is a good assumption inside a parallel-plate gap or a long solenoid and a poor one almost everywhere else.
- Add a mass if you want a specific-energy figure. Watt-hours per kilogram is the currency of energy storage comparisons, and it puts field storage into the same units as a battery.
- Read the split bar. It shows at a glance which field is doing the storing, which is usually far more lopsided than people expect.
The Formula: How Field Energy Density Is Calculated
The electric field energy density is uE = ½ε0εrE² and the magnetic energy density is uB = B² ÷ (2μ0μr). Georgia State University's HyperPhysics page on energy in electric and magnetic fields derives both from the work done in assembling the field, and shows how each reduces to the familiar total-energy expressions for a capacitor and an inductor once you multiply by the volume of the region.
The constants used here are the 2022 CODATA values. NIST gives the vacuum electric permittivity as 8.8541878188 × 10−12 farads per metre and the vacuum magnetic permeability as 1.25663706127 × 10−6 newtons per ampere squared. Since the 2019 redefinition of the SI base units neither is exact any more; both carry a small measured uncertainty inherited from the fine-structure constant, though at roughly one part in ten billion it will never limit anything on this page.
Work the defaults by hand. An electric field of 106 V/m in vacuum gives uE = 0.5 × 8.8541878 × 10−12 × 1012 = 4.427 J/m³. A flux density of 1 T gives uB = 1 ÷ (2 × 1.25663706 × 10−6) = 397,887 J/m³. The total is 397,892 J/m³, of which the magnetic term is 99.999 per cent. Over a litre of volume that is 397.9 joules — about the kinetic energy of a one-kilogram mass moving at twenty-eight metres per second.
Why Magnetic Storage Wins by Five Orders of Magnitude
The comparison above is not a quirk of the numbers chosen. It reflects a hard physical ceiling on the electric side. Electric energy density grows as the square of field strength, so pushing it up means pushing the field up — and every dielectric breaks down at some field. Dry air fails at roughly 3 × 106 V/m, which caps its energy density near 40 J/m³. Good solid dielectrics reach ten or twenty times that field and have permittivities of a few, which lifts the ceiling to the order of 105 J/m³ but no further.
Magnetic fields have no equivalent breakdown limit in the material sense. A one tesla field already stores 398 kJ/m³; the ten-tesla field of a research magnet stores a hundred times that, close to 40 MJ/m³. The limit there is mechanical rather than electrical: the magnetic pressure trying to burst the coil is numerically the same as the energy density, so a ten-tesla winding is being pushed outward at about 40 megapascals, which is roughly four hundred atmospheres.
That identity between energy density and magnetic pressure is worth holding on to. Any figure this page reports for uB in joules per cubic metre is also the magnetic pressure in pascals. It is why high-field magnets are engineering problems in structural mechanics as much as in electromagnetism, and why a superconducting magnet quench is a violent event rather than a quiet one.
The Special Case of an Electromagnetic Wave
In a plane electromagnetic wave travelling through vacuum, the electric and magnetic fields are locked together by E = cB. Put that into the two expressions and something clean falls out: the two energy densities are exactly equal at every instant. A radio wave, a light beam and a gamma ray all carry precisely half their energy in the electric field and half in the magnetic one.
This gives a useful sanity check on the calculator. Load the plane-wave preset and the split bar sits at fifty-fifty. If you type in an E and a B for what you believe is a wave in free space and the bar is lopsided, one of the two numbers is wrong — almost always because the magnetic amplitude was quoted in gauss or the field strength was an RMS value paired with a peak. Our magnetic field converter handles the first of those; the RMS voltage calculator explains the second.
The instantaneous total energy density of a plane wave is ε0E², and its time average for a sinusoidal wave is half that. Multiply the average density by the speed of light and you have the intensity in watts per square metre, which is the magnitude of the Poynting vector. Energy density and intensity are the same physical fact expressed per unit volume and per unit area.
Where the Uniform-Field Assumption Breaks
Everything on this page multiplies a density by a volume, which is only valid where the field is genuinely uniform. Inside a parallel-plate capacitor away from the edges, and inside a long solenoid away from the ends, that holds well. Almost everywhere else it does not, and the error is not small.
Around a point charge the field falls as one over distance squared, so the energy density falls as one over distance to the fourth. Nearly all the energy sits in a thin shell close to the charge, and integrating inward toward a true point diverges — a real difficulty in classical electromagnetism, not a calculation error. Around a straight wire the field falls as one over distance and the density as one over distance squared, so again the energy crowds close to the conductor.
The practical rule is that if your field varies appreciably across the region, you must integrate the density rather than multiply it. Use this tool on the largest sub-volume you can honestly call uniform, or use it to get the density at a specific point and treat the total as an order-of-magnitude figure.
Field Energy Against Other Ways of Storing It
Putting field storage next to alternatives is instructive. A one-tesla field stores about 0.11 watt-hours per litre. Compressed air at 200 bar stores roughly ten times more per litre in the ideal isothermal case. A lithium-ion cell stores several hundred watt-hours per litre. Petrol stores nearly ten thousand.
Field storage nevertheless earns its place, because energy density is not the only figure that matters. A capacitor or a magnet delivers its energy in microseconds, survives millions of cycles, and does not care about temperature the way chemistry does. Power density, not energy density, is the reason these systems exist. The same trade-off drives mechanical storage: our flywheel energy calculator works the rotating case, where the ceiling comes from material strength rather than breakdown or chemistry.
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
- Mixing gauss with tesla — one tesla is ten thousand gauss, and because the density goes as the square, a factor of ten thousand in field is a factor of a hundred million in energy.
- Multiplying a peak density by a volume for an AC field — for a sinusoidal field the time-averaged density is half the peak, so the instantaneous figure overstates the average by a factor of two.
- Treating iron's permeability as a constant — it falls sharply as the core approaches saturation, so a single μr taken from a datasheet headline can be wildly wrong at the working point.
- Applying a uniform density to a non-uniform field — around wires and point charges the energy crowds into a small region near the source, and multiplying by the whole volume overstates it badly.
- Forgetting that energy density is also magnetic pressure — the same number in pascals is the force per unit area trying to open the coil, and it is the reason high-field magnets need structural support.
Related Free Tools From Arb Digital
For the component-level view of stored electrical energy, use the capacitor energy calculator. To get the field strengths this page needs, use the electric field calculator and the solenoid magnetic field calculator. Unit problems are handled by the magnetic field converter, and AC amplitude questions by the RMS voltage calculator. For potential rather than field, see the electric potential calculator, and for the mechanical storage comparison see the flywheel energy calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the amount of energy stored per unit volume of space occupied by a field, measured in joules per cubic metre. Both electric and magnetic fields carry it, and it exists in the space itself rather than in any material object.
Yes, they add directly with no cross term. The total density is one half epsilon E squared plus B squared over two mu, and that sum is exact rather than an approximation.
Because there is no material limit on flux density in the way there is a breakdown limit on electric field. Dielectrics fail at a few tens of megavolts per metre, capping electric density, while a one tesla field already holds nearly 400 kilojoules per cubic metre.
Numerically yes. The magnetic energy density in joules per cubic metre equals the magnetic pressure in pascals, which is why a strong magnet is a structural engineering problem as much as an electrical one.
The electric and magnetic energy densities are exactly equal, because the fields are related by E equals c times B in vacuum. Half the wave energy sits in each field at every instant.
Only where the field is uniform across that volume, such as inside a parallel-plate gap or a long solenoid. Around a wire or a point charge the field varies steeply and the energy must be integrated instead.
A dielectric of relative permittivity ten multiplies the electric energy density by ten at the same field. A magnetic core changes the magnetic term through its relative permeability, but that value is strongly field-dependent and collapses near saturation.
Use whichever matches the question. Peak fields give the instantaneous maximum density; for a sinusoidal field the time-averaged density is half the peak value, and mixing the two is a common factor-of-two error.
This tool is provided for educational and study use. It assumes uniform, linear, isotropic media with constant permittivity and permeability, so treat its output as a physics teaching result rather than a design figure for magnets, capacitors or insulation systems.
