Plasma Heating Power Calculator

Ohmic heating is the plasma physics term for ordinary Joule heating: driving a current through the plasma's finite electrical resistivity dissipates power as heat, exactly as current through a resistor does in an electrical circuit. It was historically the primary heating method in early tokamaks, since the plasma current needed for confinement also provides this heating as a side effect.

P = ηJ²V, where η is the plasma's electrical resistivity, J is the current density, and V is the plasma volume. The quantity ηJ² is the local heating power density (watts per cubic metre); multiplying by volume gives the total heating power.

It computes η internally from your entered electron temperature, ion charge state, and electron density, using the same Spitzer resistivity formula (η = 1.03×10⁻⁴ Z lnΛ / T_e[eV]^1.5) as the dedicated <a href="/science/plasma-physics/spitzer-resistivity-and-conductivity-calculator/">Spitzer Resistivity and Conductivity Calculator</a>, including the NRL Coulomb logarithm computed the same way.

Because Spitzer resistivity falls as T_e^(−3/2), a hotter plasma is a better conductor, so the same current density dissipates progressively less power as heat. This is the fundamental physical reason tokamaks cannot rely on ohmic heating alone to reach fusion-relevant temperatures of 10+ keV and must add auxiliary heating methods.

The two main methods are neutral beam injection, firing high-energy neutral atoms into the plasma that transfer energy through collisions, and radiofrequency heating, which resonantly couples electromagnetic wave power into the plasma at frequencies matched to particle cyclotron or wave resonances (electron cyclotron, ion cyclotron, or lower hybrid heating).

Not directly, ohmic heating power depends on current density and resistivity, not field strength. However, the magnetic field is what confines the current-carrying plasma in the first place (via the safety factor and overall MHD equilibrium), so it plays an essential indirect role in making ohmic heating possible.

Current density J is current per unit cross-sectional area; the total plasma current is the integral of J over the plasma's cross-section. This calculator takes J directly as an input, representing either an average or local current density depending on what you are modeling.

Power density (ηJ², in W/m³) is useful for comparing local heating intensity independent of plasma size, while total power (ηJ²V, in W) is what matters for a reactor's overall heating and energy balance, so both are reported to serve either purpose.

No, since η depends on local temperature and density (and indirectly on the Coulomb logarithm), it varies across the plasma cross-section. This calculator computes a single representative value from the temperature and density you enter, appropriate for order-of-magnitude estimates rather than a fully spatially resolved calculation.

Typical tokamak ohmic heating powers range from hundreds of kilowatts to a few megawatts, depending on plasma current, size, and resistivity, generally enough to reach temperatures of roughly 1 to a few keV before auxiliary heating becomes necessary to push further toward fusion-relevant conditions.