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David J. Stevenson modeled a magnetic mechanism for hot-Jupiter radius inflation with Konstantin Batygin. Their calculation coupled winds, a planetary magnetic field and electrical conductivity to estimate how induced electric currents deposit Ohmic heating in a giant planet's interior.^[ref]

The location of the dissipated energy is crucial. Heating a thin upper atmospheric layer need not replace the deep energy lost as the planet contracts; a useful inflation model must connect atmospheric forcing to the interior heat budget.

Konstantin Batygin, with David J. Stevenson, proposed Ohmic heating of a giant planet as a mechanism for oversized hot Jupiters. Their model used the interaction between atmospheric winds and the planetary magnetic field to drive electric currents into deeper conducting material.^[ref]

In the simple resistive description, local irreversible heating is $q=J^2/\sigma$, with current density magnitude $J$ and electrical conductivity $\sigma$. Predicting a total power still requires the spatial current distribution and its closure; choosing a large wind speed alone is not a complete heating calculation.

Winds in a weakly ionized exoplanet atmosphere can drive electric currents through a planetary magnetic field. Their Ohmic heating is a proposed power source for hot-Jupiter radius inflation. A model must determine the electrical conductivity, the complete current paths and the depth at which the power is deposited.^[ref]

In a nonrelativistic, isotropically conducting fluid, let $\mathbf{v}$ be its velocity, $\mathbf{E}$ the laboratory electric field, $\mathbf{B}$ the magnetic field, $\mathbf{J}$ the current density and $\sigma >0$ the electrical conductivity. The simple resistive Ohm's law isThe irreversible heat production per unit volume and time is thereforeThe laboratory electrical power $\mathbf{J}\cdot \mathbf{E}$ alone also contains mechanical work by the Lorentz force:Thus identifying $\mathbf{J}\cdot \mathbf{E}$ directly with resistive heat requires an appropriate rest frame or a vanishing mechanical-work term. The total deposited power is $L_{\mathrm{Ohm}}=\int q_{\mathrm{Ohm}}dV$. A large value confined to optically thin upper layers need not maintain the deep interior's entropy; magnetic braking and current closure also limit what a prescribed wind can supply.