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Induced electric current in the ocean A fluid flows with uniform velocity v in the presence of a constant and uniform magnetic field B perpendicular to v. The fluid has an electrical conductivity σ. a) Find the electric current density J induced in the fluid. b) Give a numerical estimate of |J| for the terrestrial oceans, knowing that the Earth’s magnetic field has a typical value B ' 0.5 Gauss = 5 × 10−5 Tesla, the conductivity of sea water is σ ' 4 Ω−1 m−1 an a typical value of the flow velocity is v = 1 m/s. c) Due to the appearance of the induced current the magnetic force tends to slow the flow. By considering the force on a fluid element, estimate the time it would take for this effect to stop the flow, if the magnetic force only was in action. 1 Solution a) Due to the flow of the fluid, the charge carriers feel a force for unit charge equale to v × B, that is equivalent to an electric field Eeq ≡ v × B. The induced current density is J = σEeq = σv × B. (1) b) Inserting the typical values given in the text we obtain J ' 4 × 1 × 5 × 10−5 A/m2 = 2 × 10−4 A/m2 . c) For the fluid element we take a small cylinder with base surface δS and height |δl|, with δl k J. The current intensity in the cylinder is I = JδS an the force is thus given by F = Iδl × B = −BJδVv, where δV is the volume of the cylinder, which in turn has a mass m = ρδV. The equation of motion, eliminating δV and inserting J = σvB, is dv = −σB 2 v, (3) ρ dt whose solution is a decreasing exponential with a time constant τ= ρ ' 1011 s ' 3.5 × 103 yr, σB 2 being ρ = 103 kg/m3 . 2 (2) δl δS J v B (4)